\chapter{Toric Surfaces}
\label{surface}

\normalsize

\section{Singularities of Toric Surfaces and Their Resolutions}

In this chapter, we will apply most of the theory developed
so far to give a detailed treatment of the structure of
normal toric varieties of dimension two (toric surfaces).  

\subsection*{\emph{Singular Points of Toric Surfaces}}
To begin, we note the following consequences of the construction 
of normal toric varieties and the correspondence between affine
toric varieties and cones.  If $X_\Sigma$ is the toric surface 
corresponding to a fan $\Sigma$ in $N_\R \simeq \R^2$, then 
the ray generator of each of the one-dimensional cones in 
$\Sigma$ is a primitive vector (that is, it can be extended to 
a basis of $N$).  It follows from Theorem~\ref{3.fantoricprops} 
of Chapter~\ref{normal} that the toric surface obtained by 
removing the fixed points of the torus action (that is, the 
points corresponding to the two-dimensional cones under the 
Orbit-Cone Correspondence) is smooth.  There are 
only finitely many such points, so $X_\Sigma$ 
contains at most a finite number of singular points. 
Moreover, two-dimensional cones are always simplicial, so 
from Example~\ref{1.simplquot} in Chapter~\ref{affine},
each of these singular points is a finite abelian quotient
singularity (isomorphic to the image of the origin in the quotient
$\C^2/G$ where $G$ is a finite abelian group).

The following a normal form
for two-dimensional strongly convex rational polyhedral
cones in $\R^2$ will facilitate the description of
the singularities of toric surfaces.  

\begin{proposition}
\label{8.cnf}
Let $\sigma \subseteq N_\R \simeq \R^2$ be a two-dimensional strongly convex rational
polyhedral cone in $\R^2$.  Then there exists a basis $\{e_1,e_2\}$ for $N$
such that 
\[
\sigma = \Cone(e_2,me_1 - ke_2),
\]
where $m > 0$, $0 \le k < m$, and $\gcd(m,k) = 1$.  
\end{proposition}

\begin{proof}
We will need
the following statement about integer arithmetic here 
and at several other points in this chapter.

\begin{lemma}
\label{8.moddiv} 
Let $m > 0$ and $l$ be integers.  There exist unique
integers $s$ and $k$ such that 
\[
l = sm - k, \qquad 0 \le k < m.
\] 
\end{lemma}  

\noindent
The proof of the lemma is left to the reader as Exercise~\ref{8.moddivpf}.

Say $\sigma = \Cone(v_1,v_2)$ where $v_i$ are primitive
vectors.  Since $v_1$ is primitive, we can take it as part of a basis of $N$,
and we let $e_2 = v_1$.  Since $\sigma$ is strongly convex, for any 
basis $\{e_1',e_2\}$ for $N$, it will be true that
\[
v_2 = m e_1' + le_2
\]
for some $m \ne 0$.  By replacing $e_1'$ by $-e_1'$ if necessary, we
can assume $m > 0$.  By Lemma~\ref{8.moddiv}, there are integers
$s,k$ such that $l = sm - k$, where $0 \le k < m$.  Using this integer $s$, 
let $e_1 = e_1' + se_2$.  Then $\{e_1,e_2\}$ is also a basis for $N$
and 
\[
v_2 = m e_1 + (l - sm)e_2 = m e_1 - ke_2.
\]
Hence $\sigma=\Cone(e_2,me_1 - ke_2)$ as claimed.  The equation $\gcd(m,k) = 1$
follows since $v_2$ is primitive.  The uniqueness of $m,k$ is left to 
the reader as Exercise~\ref{8.uni}.
\end{proof}

We will call the integers $m,k$ in this statement the {\it parameters}
of the cone $\sigma$, and $\{e_1,e_2\}$ is called 
a {\it normalized basis} for $N$ relative to $\sigma$.  Using the normal form, we 
will next describe the local structure of the point 
$p_\sigma$ in the affine toric variety $U_\sigma$.  Recall from 
Example~\ref{1.simplquot} in Chapter~\ref{affine} that if 
$N' \subseteq N$ is the sublattice generated by the ray generators
of $\sigma$, then $U_\sigma \simeq \C^2/G$, where $G = N/N'$.
In our situation, $N = \Z e_1 \oplus \Z e_2$ and 
\[
N' = \Z e_2 \oplus \Z (m e_1 - ke_2) = m\Z e_1 \oplus \Z e_2,
\]
so it follows easily that 
\begin{equation}
\label{8.cyclic}
G = N/N' \simeq \Z/m\Z.
\end{equation}
In particular, for singularities of toric surfaces, 
the finite group $G$ is always {\it cyclic}.  

The action of $G$ on $\C^2$ is determined by the integers $m,k$
as follows.  We write 
\[
\mu_m = \{\zeta \in \C : \zeta^m = 1\},
\]
the group of $m$th roots of unity in $\C$.  Then a choice
of a primitive $m$th root of unity defines an isomorphism of 
groups $\mu_m \simeq \Z/m\Z$.


\begin{proposition}
\label{8.quotact}
Let $M'$ be the dual lattice of $N'$ and let $u_1,u_2 \in M'$
be dual to $v_1,v_2$ in $N'$.  Using the coordinates 
$x = \chi^{u_1}$ and $y = \chi^{u_2}$ on $\C^2$, the 
action of $\zeta \in \mu_m \simeq N/N'$ is given by 
\[
\zeta\cdot(x,y) = (\zeta x, \zeta^k y).
\]
\end{proposition}

\begin{proof}
From the general discussion in Chapter~\ref{affine}, $N/N'$
acts on the coordinate ring of $\C^2$ via 
\begin{equation}
\label{8.act}
(v + N')\cdot \chi^{u'} = e^{2\pi i \langle u',v\rangle} \chi^{u'},
\end{equation}
where $u' \in \sigma^\vee \cap M'$ and each element of $N/N'$
is given by $v = je_1$ for some $j$,  $0 \le j \le m - 1$.

An easy calculation shows $\langle u_1, e_1\rangle = 1/m$ 
and $\langle u_2, e_1\rangle = k/m$.  Hence if we set up 
the isomorphism $\mu_m \simeq N/N'$ by mapping
$e^{2\pi ij/m} \mapsto je_1 + N'$, then for all $\zeta = e^{2\pi ij/m} \in \mu_m$,
from \eqref{8.act}
\[
\zeta\cdot (x,y) = (e^{2\pi ij/m} x, e^{2\pi ijk/m} y) = (\zeta x, \zeta^k y),
\]
which is what we wanted to show.
\end{proof}

The isomorphism in \eqref{8.cyclic} and the result of 
Proposition~\ref{8.quotact} also indicate that there
is a slight but manageable ambiguity in the normal form 
for two-dimensional cones.  We say cones are \emph{lattice equivalent}
if there is a bijective $\Z$-linear mapping $\varphi : N \to N$
taking one cone to the other.  After choice of basis for $N$, such
mappings are defined
by matrices in $\mathrm{GL}(2,\Z)$.

\begin{proposition}
\label{8.cnfuniqueness}
Let $\sigma = \Cone(e_2, me_1 - ke_2)$ and 
$\widetilde{\sigma} = \Cone(e_2', \widetilde{m} e_1' - \widetilde{k} e_2')$ be
cones in normal form that are lattice equivalent.  Then 
$\widetilde{m} = m$ and either $\widetilde{k} = k$ or $k\widetilde{k} \equiv 1 \bmod m$.
\end{proposition}

\begin{proof}
Since the cones are lattice equivalent, writing $N'$ and $\widetilde{N}'$ for 
the sublattices as in \eqref{8.cyclic}, there is a bijective $\Z$-linear
mapping $\varphi : N \to N$ such that $\varphi(N') = \widetilde{N}'$.  Hence
$N/\widetilde{N}' \simeq N/N'$, so $\widetilde{m} = m$.  The
statement about $k$ and $\widetilde{k}$ is left to the reader in Exercise~\ref{8.uni}.
\end{proof}

We will next consider two examples to illustrate Proposition~\ref{8.quotact}
and we will identify the resulting toric surface singularities.  

\begin{example}
\label{8.rncagain}
First consider a cone 
\[
\sigma = \Cone(e_2,me_1 - e_2)
\]
with parameters $m > 1$ (so the cone is not smooth) and $k = 1$.  
Apart from slightly different notation, this is precisely the cone
considered in Example~\ref{1.rnccone2} from Chapter~\ref{affine}.
The corresponding toric surface $U_\sigma$ is the 
cone over the rational normal curve of degree $m$: 
$\widehat{C}_m \subseteq \C^{m+1}$.  The quotient structure
was studied in the special case $m = 2$ in Example~\ref{1.redornc2ex}
from Chapter~\ref{affine}, and the general case
was described in Exercise~\ref{1.3.rncexer}.  With the notation
of Proposition~\ref{8.quotact}, $\zeta \in \mu_m$
acts on $(x,y) \in \C^2$ via 
$\zeta\cdot (x,y) = (\zeta x, \zeta y)$ and
the ring of invariants is 
\[
\C[x,y]^{\mu_m} = \C[x^m, x^{m-1}y, \ldots, xy^{m-1}, y^m].
\]
so 
\[
U_\sigma \simeq \C^2/\mu_m \simeq \Spec(\C[x^m,x^{m-1}y,\ldots,xy^{m-1}, y^m]).
\]
On the other hand, from the Hilbert basis of the semigroup
$\sS_\sigma = \sigma^\vee \cap M$ we obtain the standard
toric description
\[
U_\sigma \simeq \Spec(\C[u,uv,uv^2,\ldots,uv^m]).
\]
Exercise~\ref{8.rnccomp} studies the relation between 
these two representations of the coordinate ring of $U_\sigma$.  
\eoe
\end{example}

\begin{example}
\label{8.rdp}
Next consider a cone $\sigma$ with parameters $m$ and $k = m - 1$,
so $m = k + 1$.  (We will express everything in terms of the 
parameter $k$ in the following.)
Unlike the previous example, this is a case we have not 
encountered previously.  Note that $k \equiv -1 \bmod m$.  Hence
by Proposition~\ref{8.quotact}, the action of $G = N/N'$
on $\C^2$ is given by 
\[
\zeta\cdot(x,y) = (\zeta x, \zeta^{-1}y).
\]
It is easy to check that the ring of invariants here is 
\[
\C[x,y]^{\mu_{k+1}} = \C[x^{k+1},y^{k+1},xy].
\]
Moreover we have an isomorphism of rings
\begin{eqnarray*}
\varphi : \C[X,Y,Z]/\langle Z^{k+1} - XY\rangle & \simeq & \C[x^{k+1},y^{k+1},xy]\\
      X & \mapsto & x^{k+1}\\
      Y & \mapsto & y^{k+1}\\
      Z & \mapsto & xy,
\end{eqnarray*}
so we may identify the toric surface $U_\sigma$ with the 
variety $\V(Z^{k+1} - XY) \subseteq \C^3$.

This variety has a singular point at the origin 
called a \emph{rational double point} (or Du Val singularity)
of type $A_k$.  Another standard
form of these singularities is given in Exercise~\ref{8.altrdp}.   
They are called \emph{double} points because the lowest degree nonzero 
term in the defining equation has degree two (that is, the
\emph{multiplicity} of the singularity is two) for all $k$. 
The \emph{rational} double points are the simplest singularities
from a certain point of view.  The exact definition, which 
we will give in \S 3 of this chapter, depends on
the notion of a resolution of singularities, which will be
introduced shortly.  All rational double
points appear as singularities of quotient surfaces $\C^2/G$ 
where $G$ is a finite subgroup of ${\rm SU}(2,\C)$.  There is a 
complete classification of such points in terms of the \emph{Dynkin 
diagrams} of types $A_k, D_k, E_6, E_7$, and $E_8$.   The groups 
corresponding to the diagrams $D_k, E_6, E_7, E_8$ are
not abelian, however, so by Example~\ref{1.simplquot} in 
Chapter~\ref{affine}, such points do not appear on toric
surfaces.  We will see one way that the Dynkin diagram $A_k$ appears 
from the geometry of the toric surface $U_\sigma$ in Exercise~\ref{8.rdpcheck}, 
and we will return to this example in \S 3 of this chapter.
More details on these singularities can be found in 
\cite[Chapter VI]{shafarevich} and in the article
\cite{durfee}.
\eoe
\end{example}

\subsection*{\emph{Resolutions of Singularities}}
A morphism of irreducible varieties $F : Y \to X$
induces a mapping $F^* : \C(X) \to \C(Y)$ on function
fields.  We say $F$ is a \emph{birational} morphism  
if $F^*$ is an isomorphism of fields.  It is a standard
consequence of this condition that there are nonempty 
Zariski-open subsets $U \subseteq Y$ and $V \subseteq X$
such that $F|_U$ gives an isomorphism from $U$ to $V$. (A proof 
of a more general statement that implies this is sketched in 
Exercise~\ref{8.isosubsets}.)  However
$F$ can fail to be injective on $Y \setminus U$.   
Important examples of birational morphisms are the 
standard blowups
\begin{equation}
\label{8.blup}
\varphi : \mathrm{Bl}_0(\C^n) \to \C^n
\end{equation}
and the toric blowup morphisms 
\[
\varphi : X_{\Sigma^*(\sigma)} \to X_\Sigma
\]
generalizing \eqref{8.blup} 
constructed in Proposition~\ref{3.starblowup} from
Chapter~\ref{normal}.  In the
case of the blowup of the origin in $\C^n$, the open sets 
$V = \C^n \setminus \{0\}$ and 
$U = \mathrm{Bl}_0(\C^n) \setminus \varphi^{-1}(\{0\})$
are isomorphic, while $\varphi^{-1}(\{0\}) \simeq \P^{n-1}$.

In the basic example $\varphi : \mathrm{Bl}_0(\C^n) \to \C^n$,
$\C^n$ is itself a smooth variety.  However, 
blowup mappings are often used as follows. 
Starting from a singular variety $X$ (such as a toric 
variety $X_\Sigma$ with singular points), we may attempt
to find a blowup (or perhaps the composition of a sequence of blowups) 
\[
\pi : Y \to X
\]
such that the complexity of the singularities of $Y$ is 
reduced in some suitable sense.  If we can remove or ``resolve'' 
the singularities entirely, so the end result $Y$ is smooth, 
then we have obtained a smooth variety with the 
same function field as $X$.  Intuitively, 
we might think of $Y$ as a desingularized, but birationally 
equivalent, version of the original
$X$.  This motivates the following definition.

\begin{definition}
\label{8.ressings}
Let $X$ be a singular variety.  A \emph{\bfseries resolution of 
singularities} of $X$ is a smooth variety $Y$ and a birational proper 
morphism $\pi : Y \to X$.
\end{definition}

The existence of resolutions of singularities for curves was 
established in the 19th century.  Resolutions of singularities
for surfaces and threefolds over $\C$ were established by Zariski in 1939 and 1944, 
respectively.  The existence of a resolution of singularities
for all $X$ over $\C$ was proved in the landmark 1964 article
\cite{hironaka} of Hironaka.  We note that one direction of research 
in modern algebraic geometry seeks to extend results such as this
to the case of varieties over general fields, including fields
of characteristic $p$.  Unfortunately, Hironaka's general approach 
does not work in characteristic $p$.  While existence of resolutions 
for curves in this case follows by the classical approaches and
the result for surfaces and some threefolds was proved by Abhyankar, 
the existence of resolutions of singularities in characteristic $p$ is still 
an open problem for varieties of dimension $\ge 4$ and for threefolds 
if $p \le 5$.

One of the interesting and appealing aspects of toric varieties
is the way that many questions that are extremely difficult for general
varieties admit beautifully simple and concrete solutions in the toric case.  
The problem of finding resolutions of singularities is a 
perfect example.  Let us begin to illustrate this by constructing 
explicit resolutions of the toric surface singularities from 
Example~\ref{8.rncagain} and Example~\ref{8.rdp}.

\begin{example}
\label{8.rnccont}
Consider the rational normal cone
of degree $m$ -- the affine toric surface 
$U_\sigma$ for for $\sigma = \Cone(e_2, me_1 - e_2)$ 
studied in Example~\ref{8.rncagain}.  Let $\Sigma'$ be 
the fan consisting of $\sigma$ and its faces and let $\Sigma$
be the fan in Figure~1 obtained by inserting
a new ray $\tau = \Cone(e_1)$ subdividing $\sigma$
into two two-dimensional cones:
\begin{eqnarray*}
\sigma_1 & = & \Cone(e_2,e_1)\\
\sigma_2 & = & \Cone(e_1,me_1 - e_2).
\end{eqnarray*}

\begin{figure}
\begin{center}
\includegraphics{Chapter8Fig1.ps}
\end{center}
\caption{The cones $\sigma_1,\sigma_2,\tau$ in Example~\ref{8.rnccont}} 
\end{figure}

We will now make use of a number of the results from 
Chapter~\ref{normal} and all references will be to that chapter. 
First, the identity mapping on the 
lattice $N$ is compatible with the fans $\Sigma$ and $\Sigma'$ as
in Definition~\ref{3.compatiblemap}.
By Theorem~\ref{3.mapcorresp}, we have a 
corresponding toric blowup morphism
\begin{equation}
\label{8.tresrnc}
\pi : X_\Sigma \to X_{\Sigma'} = U_\sigma.
\end{equation}
Note that both $\sigma_1$ and $\sigma_2$ (as well as all of 
their faces) are smooth cones.  Hence Theorem~\ref{3.fantoricprops}
implies that $X_\Sigma$ is a smooth surface.  In addition, 
the toric morphism $\pi$ is proper by Theorem~\ref{3.propercriterion}
since $\Sigma$ is a refinement of $\Sigma'$.  Finally, we claim
that $\pi$ is birational.  This follows automatically in this
case because $\pi$ restricts to the identity mapping from
the torus $T_N \subseteq X_\Sigma$ to $T_N \subseteq X_{\Sigma'}$
and these are Zariski-open subsets of the surfaces in question.  But
of course, we can say more.  By the Orbit-Cone
Correspondence, it follows that if $p_\sigma$ is the
distinguished point corresponding to the two-dimensional
cone $\sigma$ in $\Sigma'$ (the singular point
of $U_\sigma$ at the origin), then $\pi$ restricts to an isomorphism
\[
X_\Sigma \setminus \pi^{-1}( \{ p_\sigma \} ) \simeq X_{\Sigma'} \setminus \{p_\sigma\}.
\]
The inverse image $E = \pi^{-1}(\{p_\sigma\})$ is the curve on $X_\Sigma$
given by the closure of the $T_N$-orbit $O(\tau)$ corresponding to 
the ray $\tau$.   That is, the singular point 
``blows up'' to $E \simeq \P^1$ on the smooth surface.   It follows
that $X_\Sigma$ and the morphism \eqref{8.tresrnc}
give a toric resolution of singularities of the rational normal 
cone.  $E$ is called the \emph{exceptional divisor} on the smooth
surface.  We will study more of the geometry of how $E$ sits inside
the surface $X_\Sigma$ in \S 3.
\eoe
\end{example}

\begin{example}
\label{8.rdpcont}
We consider the case $m = 4$ of Example~\ref{8.rdp}, for which
the surface $U_\sigma$ has a rational double point of type $A_3$.  
We will leave the verification of our claims, as well as the 
generalization to all $m \ge 2$ to the reader in 
Exercise~\ref{8.rdpcheck}.  As in the previous example, write
$\Sigma'$ for the fan consisting of the cone 
\[
\sigma = \Cone(e_2, 4e_1 - 3e_2)
\]
and its faces.  It is easy to find subdivisions of 
$\sigma$ yielding collections of smooth cones.  The most
economical way to do this is to insert three new
rays $\rho_1 = \Cone(e_1)$, $\rho_2 = \Cone(2e_1 - e_2)$,
$\rho_3 = \Cone(3e_1 - 2e_2)$
to obtain a fan $\Sigma$ consisting of four two-dimensional cones
and their faces. 

\begin{figure}
\begin{center}
\includegraphics{Chapter8Fig2.ps}
\end{center}
\caption{The cone $\sigma$ and the refinement $\Sigma$ in Example~\ref{8.rdpcont}} 
\end{figure}

The process is perhaps even clearer if we draw the cones
relative to a different basis $\{f_1, f_2\}$ for $N$.  Taking
$f_1 = e_2$ and $f_2 = e_1 - e_2$, the subdivided fan 
consisting of
\begin{eqnarray}
\label{8.rdpcones}
\sigma_1 & = & \Cone(f_1, f_1 + f_2)\nonumber\\
\sigma_2 & = & \Cone(f_1 + f_2, f_1 + 2 f_2)\\
\sigma_3 & = & \Cone(f_1 + 2 f_2, f_1 + 3 f_2), \nonumber\\
\sigma_4 & = & \Cone(f_1 + 3 f_2, f_1 + 4 f_2) \nonumber
\end{eqnarray}
and their faces appears in Figure 2.  


You will check that each of these cones is smooth.  
Hence $X_\Sigma$ is a smooth surface.  As in the previous
example, since $\Sigma$ is a refinement of $\Sigma'$, we
have a toric blowup that yields a resolution of singularities
\[
\pi : X_\Sigma \to X_{\Sigma'} = U_\sigma.
\]
Also as in the previous example, $\pi$ restricts
to an isomorphism from $X_\Sigma \setminus \pi^{-1}(\{p_\sigma\})$ 
to $X_{\Sigma'} \setminus \{p_\sigma\}$.  In this case,
the exceptional divisor 
$E = \pi^{-1}(\{p_\sigma\})$ is the union
\[
E = \overline{O(\rho_1)} \cup \overline{O(\rho_2)} \cup \overline{O(\rho_3)}
\]
on $X_\Sigma$.  Each of the curves $\overline{O(\rho_i)}$
is isomorphic to $\P^1$.  The first two intersect transversely
at the fixed point of the $T_N$-action on $X_\Sigma$ 
corresponding to the cone $\sigma_2$, while the second two
intersect transversely at the fixed point corresponding to $\sigma_3$.
\eoe
\end{example}

In these examples, we constructed resolutions 
of affine toric surfaces with just one singular point.
The same techniques can be applied to any normal
toric surface $X_{\Sigma'}$.

\begin{theorem}
\label{8.toricressings}
Let $X_{\Sigma'}$ be a normal toric surface.
There exists a smooth fan $\Sigma$ refining $\Sigma'$
such that the associated toric morphism
$\pi : X_\Sigma \to X_{\Sigma'}$
is a resolution of singularities.
\end{theorem}

\begin{proof}
If we can show the existence of the smooth fan
$\Sigma$ refining $\Sigma'$, the reasoning given in Example~\ref{8.rnccont} applies
to show that the corresponding toric morphism $\pi$ is proper
and birational, hence a resolution of singularities of $X_{\Sigma'}$.
Hence, we only need to show that $\Sigma$ exists.  

We will prove this by induction on an integer invariant
of fans that measures the complexity of the singularities
on the corresponding surfaces.  Let $\sigma_1, \ldots, \sigma_n$
denote the two-dimensional cones in a fan $\Delta$.  For each $i$, 
we will write $N_i$ for the sublattice of $N$ generated
by the ray generators of $\sigma_i$.  Then we define
\[
s(\Delta) = \sum_{i=1}^n \left([N:N_i] - 1\right).
\]
If $s(\Delta) = 0$, then $n = 0$ or $[N:N_i] = 1$ for all $i$.
It is easy to see that this implies that $\Delta$ is 
a smooth fan.  Hence $X_\Delta$
is already smooth and we take this as the base case for 
our induction.  

For the induction step, we assume that the existence
of smooth refinements has been established for all 
fans $\Delta$ with $s(\Delta) < s$, and consider
a fan $\Delta'$ with $s(\Delta') = s$.   If $s \ge 1$,
then there exists some nonsmooth cone $\sigma_i$ in 
$\Delta'$.  By Proposition~\ref{8.cnf}, there is a basis
$\{e_1, e_2\}$ for $N$ such that 
$\sigma_i = \Cone(e_2,me_1 - ke_2)$
with parameters $m > 0$, $0 \le k < m$, and $\gcd(m,k) = 1$.
Consider the refinement $\Delta$ of $\Delta'$ obtained
by subdividing the cone $\sigma_i$ into two new 
cones 
\begin{eqnarray*}
\sigma_i' & = & \Cone(e_2,e_1)\\
\sigma_i'' & = & \Cone(e_1,me_1 - ke_2)
\end{eqnarray*}
with a new one-dimensional cone $\rho = \Cone(e_1)$.
We must show that $s(\Delta) < s(\Delta')$ to invoke
the induction hypothesis and conclude the proof.  

In $s(\Delta)$, the terms corresponding to the other
cones $\sigma_j$
$j \ne i$ from $\Delta'$ are unchanged.  The cone 
$\sigma_i'$ is smooth since $\{e_1,e_2\}$ is the normalized
basis of $N$ relative to $\sigma_i$.   So it contributes a zero term
in $s(\Delta)$.  Finally, we consider the cone
$\sigma_i''$.  We saw in \eqref{8.cyclic} that
if $\sigma$ is a cone with parameters $m,k$
then the sublattice $N'$ generated by the 
ray generators satisfies $[N:N'] = m$.  Hence we must
determine the parameters of the cone
$\sigma_i''$.  

In terms of the basis $\{e_1,e_2\}$
for $N$, the $\Z$-linear mapping defined by the matrix
\[
A = \begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix}
\]
(a ``90-degree rotation'') takes $\sigma_i''$
to $\Cone(e_2,ke_1 + me_2)$.  Since $A \in \mathrm{GL}(2,\Z)$,
it defines an automorphism of $N$, and hence $\sigma_i''$
will have the same parameters as  $\Cone(e_2,ke_1 + me_2)$.
But now we apply Lemma~\ref{8.moddiv} to put this cone
into normal form.  There is some integer $l$, $0 \le l < k$,
such that our cone has parameters $k, l$.  Since
$k < m$, if $N_i''$ is the sublattice 
generated by the ray generators of $\sigma_i''$, then
by \eqref{8.cyclic},
\[
[N:N_i''] = k < [N:N_i] = m.
\]
It follows that 
\[
s(\Delta) < s(\Delta'),
\]
and the proof is complete by induction.
\end{proof}

We will see in the next section that the one-dimensional
cones in the refined fan $\Sigma$ yielding the resolution
of $X_\Sigma'$  have a very nice description.  Although to 
this point we have usually only considered the semigroups
$\sigma^\vee \cap M$ defined by the dual cones, we can also 
consider $\sigma \cap N$ for $\sigma$ themselves.  For each
two-dimensional cone $\sigma$ in $\Sigma'$, the new cones 
in $\Sigma$ are produced by subdividing along the rays through
the irreducible elements of the semigroup $\sigma \cap N$.  
If you look back at Examples~\ref{8.rnccont}
and \ref{8.rdpcont}, you will see how this works in two special
cases.  

A resolution of a non-normal toric surface singularity
can be constructed by first saturating the associated
semigroup as in Theorem~\ref{1.satthm} from Chapter~\ref{affine}, 
then applying the results of this section.  Similar techniques can also be applied 
to construct toric blowups giving resolutions of singularities 
of toric varieties of dimension three and larger.  
However, we will postpone discussing the higher-dimensional 
case until Chapter~\ref{resolution}.

\exercises

\small

\begin{exercise}
\label{8.moddivpf}
Adapt the usual proof of the integer division algorithm
to prove Lemma~\ref{8.moddiv}.
\end{exercise}

\begin{exercise}
\label{8.uni}  In this exercise, you will develop further 
properties of the parameters $m,k$ in the normal form for cones from 
Proposition~\ref{8.cnf} and prove part of Proposition~\ref{8.cnfuniqueness}.
\begin{thmenum}
\item Show that if $\widetilde{\sigma}$ is obtained from a
cone $\sigma$ by parameters $m,k$ by 
a $\Z$-linear mapping of $N$ defined by a matrix 
in $\mathrm{GL}(2,\Z)$, then the parameter $\widetilde{k}$
of $\widetilde{\sigma}$ satisfies either $\widetilde{k} = k$,
or $k\widetilde{k} \equiv 1 \bmod m$.  Hint: there is a 
choice of \emph{orientation} to be made in the normalization process.
Recall that $\gcd(m,k) = 1$ so there are integers $\widetilde{m},\widetilde{k}$
such that $\widetilde{m} m + \widetilde{k} k = 1$.
\item Show directly using Proposition~\ref{8.quotact}
that the singularities at the origin of the toric varieties
$U_\sigma$ and $U_{\widetilde{\sigma}}$ are isomorphic
if $\sigma$ has parameters $m,k$ and $\widetilde{\sigma}$
has parameters $m, \widetilde{k}$ with 
$k\widetilde{k} \equiv 1 \bmod m$. 
\item Show that if $\sigma$ is a cone
with parameters $m,k$, then the dual cone
$\sigma^\vee \subseteq M_\R$ has parameters $m, m - k$.
Hint:  Use the normal form for $\sigma$, write down
$\sigma^\vee$ in the corresponding dual coordinates in $M$, 
then change coordinates in $M$ to normalize $\sigma^\vee$.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{8.rnccomp}
With the notation in Example~\ref{8.rncagain}, show that
\[
\C[u,uv,uv^2,\ldots, uv^m] \simeq \C[x^m, x^{m-1}y, \ldots, xy^{m-1}, y^m]
\]
under $u \mapsto x^m$ and $v \mapsto y/x$, and use Proposition~\ref{8.quotact}
to explain where
these identifications come from in terms of the semigroup $\sS_\sigma$.
Hint: We have $u = \chi^{e_1^*}$ and $v = \chi^{e_2^*}$
where $\{e_1,e_2\}$ is the normalized basis for $N$ and $\{e_1^*,e_2^*\}$
is the dual basis for $M$.
\end{exercise}

\begin{exercise}
\label{8.altrdp}  In Example~\ref{8.rdp}, we gave one form
of the rational double point of type $A_k$, namely the singular
point at $(0,0,0)$ on the surface $V = \V(Z^{k+1} - XY) \subseteq \C^3$.  Another
commonly used normal form for this type of singularity is the 
singular point at $(0,0,0)$ on the surface $W = \V(X^{k+1} + Y^2 + Z^2)$.
Show that $V$ and $W$ are isomorphic as affine varieties, hence
the singularities at the origin are analytically equivalent.  Hint: there
is a linear change of coordinates in $\C^3$ that shows this. 
\end{exercise}

\begin{exercise}
\label{8.isosubsets}
Let $X,Y$ be irreducible varieties.  Consider the class
of morphisms $F : U \to Y$ where $U$ is a nonempty Zariski-open
subset of $X$.  We will say $F : U \to Y$ and $F' : V \to Y$
are equivalent if $F\res{U\cap V} = F'\res{U\cap V}$.  
A \emph{rational map} $F : X \smallda Y$ is by definition an
equivalence class of such morphisms under this notion of equivalence.  
A rational mapping
$F : X \smallda Y$ is \emph{birational} if it has a rational inverse 
$G : Y \smallda X$ (that is $F \circ G = id$ and $G \circ F = id$
where the compositions are defined).  
\begin{thmenum}
\item Show that a birational mapping $F$ induces an isomorphism
of function fields $F^* : \C(Y) \simeq \C(X)$.
\item Show that if $F$ is birational, then there are nonempty Zariski-open subsets 
$U \subseteq X$ and $V \subseteq Y$ such that $F$ restricts
to an isomorphism from $U$ to $V$.  (This applies in particular
when $F$ is a birational morphism, hence a mapping with
domain equal to $X$.)
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{8.rdpcheck}
In this exercise, you will check several of the claims made in 
Example~\ref{8.rdpcont} and show how to extend the results there
to the case $\sigma = \Cone(e_2,me_1 - (m-1)e_2)$ for general $m$. 
\begin{thmenum}
\item Check that each of the four cones in \eqref{8.rdpcones}
is smooth, so that the toric surface $X_\Sigma$ is smooth
by Theorem~\ref{3.fantoricprops} from Chapter~\ref{normal}.
\item For general $m$, show how to insert new rays $\rho_i$ 
to subdivide $\sigma$ and obtain a fan $\Sigma$ whose associated
toric surface is smooth.  Try to do this with as few new 
rays as possible.  Hence we obtain toric resolutions
of singularities $\pi : X_\Sigma \to U_\sigma$ for all $m$.
\item Identify the inverse image $C = \pi^{-1}(\{p_\sigma\})$ 
in general.  For instance, how many irreducible components does $C$ have?
How are they connected?  Hint:  One way to represent the structure is 
to draw a graph with vertices corresponding to the components
and to connect two vertices by an edge if and only if the components
intersect on $X_\Sigma$.  Do you notice a relation between 
this graph and the Dynkin diagram $A_k = A_{m-1}$ mentioned before?
We will discuss the relation in detail in \S 3 of this chapter.  
\end{thmenum}
\end{exercise}

\normalsize
\section{Hirzebruch-Jung Continued Fractions}

Let $\sigma \subseteq N_\R$ be a cone with parameters $m,k$
as in Proposition~\ref{8.cnf}.  
The process of constructing the resolution of singularities
for $U_\sigma$ as in Theorem~\ref{8.toricressings}
has an interesting interpretation in terms
of a modification of the \emph{continued fraction expansion} 
of a rational number from elementary number theory.   We will
define a variant known as the Hirzebruch-Jung continued fraction
in this section, and show how the operations from \S 1 can be interpreted
in terms of these continued fractions.  We will also use them to draw some 
additional conclusions about the sequences of toric blowups involved
in resolutions of toric surface singularities. 

\subsection*{\emph{\bfseries Ordinary Continued Fractions}}
If $m > k > 0$ are integers, then the ordinary continued
fraction expansion of $m/k$ may be obtained by performing the
same sequence of integer divisions employed in the Euclidean
algorithm for the gcd.  Starting with $k_{-1} = m$ and $k_0 = k$,
we write $a_i$ for the quotient
and $k_i$ for the remainder at each step, so that the $i$th 
division is given by 
\begin{equation}
\label{8.ordEuc}
k_{i-2} = a_i k_{i-1} + k_i,
\end{equation}
where $0 \le k_i < k_{i-1}$.  

Say $k_{n-1}$ is the final nonzero remainder (which equals $\gcd(m,k)$).
The resulting equations can be rearranged:
\begin{eqnarray}
\label{8.cfraccomps}
    m/k & = & a_1 + k_1/k\nonumber\\ 
  k/k_1 & = & a_2 + k_2/k_1\nonumber\\
        & \vdots & \\
k_{n-3}/k_{n-2} & = & a_{n-1} + k_{n-1}/k_{n-2}\nonumber\\
k_{n-2}/k_{n-1} & = & a_n,\nonumber
\end{eqnarray}
and spliced together to form the continued fraction
\[
m/k = a_1 + \frac{1}{a_2 + \frac{1}{ \cdots\ + \frac{1}{a_n} }}.
\]

For obvious typographical reasons, it is desirable to have a more compact
way to represent these expressions.  We will use the notation
\begin{equation}
\label{8.cfrac}
m/k = [a_1, a_2, \ldots, a_n]
\end{equation}
for the ordinary continued fraction.  In the context of continued
fraction expansions, the $a_i$ are
called the \emph{partial quotients} of $m/k$; the truncated continued
fractions 
\[
c_i = [a_1, a_2, \ldots, a_i],  \quad 1 \le i \le n
\]
are called the \emph{convergents} of $m/k$.  For future reference, 
we mention that there are recurrences for the numerators
and denominators of the convergents. 

\begin{proposition}
\label{8.cfracconvprop}
Let $m > k > 0$ be integers.  
Define sequences $p_i$ and $q_i$ recursively as follows.
First set
\begin{eqnarray*}
p_0 = 1 & \qquad & q_0 = 0\\
p_1 = a_1 & \qquad & q_1 = 1.
\end{eqnarray*} 
Then, for all $2\le i \le n$, let 
\begin{eqnarray}
\label{8.ordcfracrecurr}
p_i & = & p_{i-2} + a_i p_{i-1}\\
q_i & = & q_{i-2} + a_i q_{i-1}.\nonumber
\end{eqnarray}
The convergents of the ordinary continued
fraction expansion for $m/k$ are given by
\[
c_i = p_i/q_i, \quad 1 \le i \le n. 
\]
Moreover, $\gcd(p_i,q_i) = 1$ for all $i$.
\end{proposition}

The proof is left to the reader as Exercise~\ref{8.ordconvex}.
Ordinary continued fraction expansions can also be constructed for
arbitary real numbers $\alpha$ by modifying \eqref{8.cfraccomps}
appropriately.  If $\alpha$ is irrational, the expansion will be 
infinite (see Exercise~\ref{8.arbcfrac}). 

\subsection*{\emph{Hirzebruch-Jung Continued Fractions}}  What
do continued fractions have to do with toric surfaces?  To answer
this, let us consider what happens when we start with the singular
point at the origin on the affine toric surface $U_\sigma$ for
a cone $\sigma$ in normal form with parameters $m,k$, and construct
a resolution of singularities using a process following the 
proof of Theorem~\ref{8.toricressings}.
The first step in the resolution is to refine the cone $\sigma$
to a fan $\Sigma$ containing the two-dimensional cones
\[
\sigma' = \Cone(e_2,e_1) \quad \hbox{and} \quad \sigma'' = \Cone(e_1,me_1 - ke_2).
\]
The first is smooth, but the second may not be.  However, we saw
that the cone $\sigma''$ has parameters $k, k_1$ where
\[
m = b_1 k - k_1,
\]
$b_1 > 0$, $0 \le k_1 < k$ (as in Lemma~\ref{8.moddiv}).  We used slightly
different notation before, writing $s$ rather than $b_1$.  The
new notation will help us keep track of what happens if we
now continue the process and refine the cone $\sigma''$.  

Just as before, using the normalized basis for $N$ relative to $\sigma''$ we
insert a new ray and obtain a new smooth cone and a second, possibly
nonsmooth cone with parameters $k_1, k_2$, where 
\[
k = b_2 k_1 - k_2
\]
using Lemma~\ref{8.moddiv}.

This lemma is a sort of modified division algorithm, and 
applying it repeatedly yields a modified Euclidean algorithm
\begin{eqnarray}
\label{8.modEuc}
m & = & b_1 k - k_1 \nonumber\\
k & = & b_2 k_1 - k_2\nonumber\\
    & \vdots & \\
k_{r-3} & = & b_{r-1} k_{r-2} - k_{r-1}\nonumber\\
k_{r-2} & = & b_r k_{r-1}\nonumber
\end{eqnarray}
that computes the parameters of the new cones produced
as we successively subdivide to produce the fan
giving the resolution of singularities.
The process terminates with $k_r = 0$ for some $r$ as shown,
since as in the usual Euclidean algorithm, 
the $k_i$ are a strictly decreasing sequence of nonnegative
numbers.  Since $0\le k_{i+1} < k_i$ for all $i$
by Lemma~\ref{8.moddiv}, we will
have $b_i \ge 2$ for all $i$.  

Now, rearranging these equations just as we did before
in \eqref{8.cfraccomps} gives a new type of continued
fraction expansion for the rational number $m/k$, with
minus signs replacing the plus signs:
\begin{equation}
\label{8.HJcfrac}
m/k = b_1 - \frac{1}{b_2 - \frac{1}{ \cdots\ - \frac{1}{b_r}  }}.
\end{equation}
This is called the \emph{Hirzebruch-Jung continued fraction 
expansion}.  We will abbreviate it as
\[
m/k = [[b_1, b_2, \ldots, b_r]].
\]
The integers $b_i$ are called the \emph{partial quotients}
of the Hirzebruch-Jung continued fraction.  The truncated Hirzebruch-Jung
continued fractions:
\[
d_i = [[b_1, b_2, \ldots, b_i]], \quad 1 \le i \le r
\]
are the \emph{convergents}.  We have $d_r = [[b_1, b_2, \ldots, b_r]] = m/k$.

\begin{example}
\label{8.cfcompare}
Consider the rational number $17/11$.  The ordinary continued
fraction expansion is
\[
17/11 = [1, 1, 1, 5].
\]
On the other hand, the Hirzebruch-Jung continued fraction
expansion is
\[
17/11 = [[2, 3, 2, 2, 2, 2]].
\]
Note that the partial quotients and the lengths are both
different.  As noted above, 
all $b_i \ge 2$ in a Hirzebruch-Jung continued fraction.
Each of these expansions determines the other and there
are methods for computing the Hirzebruch-Jung partial
quotients $b_j$ in terms of the ordinary partial quotients $a_i$.
See, for instance, \cite[p. 257]{hirzebruch} and Exercise~\ref{8.convert} 
below.
\eoe
\end{example}

\subsection*{\emph{\bfseries Hirzebruch-Jung Continued Fractions and Resolutions}}
The process of computing the Hirzebruch-Jung continued
fraction of $m/k$ by the modified Euclidean algorithm
\eqref{8.modEuc} also yields a very convenient
computational method for producing the fan $\Sigma$ such that 
$\pi : X_\Sigma \to U_\sigma$ is a resolution of singularities. 
The following process for 
computing the numerator and denominators of the $d_i$
is the direct generalization of Proposition~\ref{8.cfracconvprop}.

\begin{proposition}
\label{8.HJconvprop}
Let $m > k > 0$ be integers.  
Define sequences $p_i$ and $q_i$ recursively as follows.
First set
\begin{eqnarray*}
  p_0 = 1 & \qquad & q_0 = 0 \\
p_1 = b_1 & \qquad & q_1 = 1.
\end{eqnarray*} 
Then, for all $2\le i \le r$, let 
\begin{eqnarray}
\label{8.HJrecurr}
p_i & = & -p_{i-2} + b_i p_{i-1}\\
q_i & = & -q_{i-2} + b_i q_{i-1}.\nonumber
\end{eqnarray}
The convergents of the Hirzebruch-Jung continued
fraction expansion for $m/k$ are given by
\[
d_i = p_i/q_i, \quad 1 \le i \le r. 
\]
\end{proposition}

There are two differences between the recurrences in \eqref{8.HJrecurr}
and those in \eqref{8.ordcfracrecurr}.  First, note the minus signs on 
the right of each equation.  Second, the $b_i$ from the Hirzebruch-Jung
continued fraction are used in place of the $a_i$ from the ordinary
continued fraction.  You will adapt the proof
of Proposition~\ref{8.cfracconvprop} to prove this in 
Exercise~\ref{8.HJconvex}.  We next show how to compute
the fan $\Sigma$ refining the cone $\sigma$ using this information.

\begin{theorem}
\label{8.HJres}
Let $\sigma$ be a cone with parameters $m,k$ and let 
$\{e_1,e_2\}$ be the normalized basis for $N$ relative to $\sigma$.
Let $v_0 = e_2$.  Using the integers $p_i$ and $q_i$ from 
Proposition~\ref{8.HJconvprop}, construct vectors
\[
v_i = p_{i-1} e_1 - q_{i-1} e_2, \quad 1 \le i \le r+1,
\]
and cones
\[
\sigma_i = \Cone(v_{i-1},v_i), \quad 1 \le i \le r+1.
\]
Then
\begin{thmenum}
\item Each $\sigma_i$ is a smooth cone and $v_{i-1},v_i$ are 
its ray generators.
\item For each $i$, $\sigma_{i+1} \cap \sigma_i = \Cone(v_i)$.
\item We have $\sigma_1 \cup \cdots \cup \sigma_{r+1} = \sigma$
so the fan $\Sigma$ consisting of the $\sigma_i$ and their faces gives
a refinement of $\sigma$.  
\item The toric morphism $\pi : X_\Sigma \to U_\sigma$ is a resolution
of singularities.
\end{thmenum}
\end{theorem}

\begin{proof}  
Both statements in part a will follow from the equations
\begin{equation}
\label{8.deteq}
p_{i-1}q_i - p_iq_{i-1} = 1, \quad 1\le i \le r.
\end{equation}
We will prove this by induction on $i$.  The base case $i = 1$
follows directly from the initialization of the $p_i,q_i$ sequences
in Proposition~\ref{8.HJconvprop}.  Now assume that the result has
been proved for $i \le s$, and consider $i = s+1$.  Using the 
recurrences \eqref{8.HJrecurr} we have
\begin{eqnarray*}
p_sq_{s+1} - p_{s+1}q_s & = & p_s(-q_{s-1} + b_s q_s) - (-p_{s-1} + b_s p_s)q_s\\
& = & p_{s-1}q_s - p_sq_{s-1} \\
& = & 1
\end{eqnarray*}
by the induction hypothesis.  The fact that $v_i$ are 
primitive follows from \eqref{8.deteq}, since these equations
show $\gcd(p_i,q_i) = 1$ for all $i$.  

For part b, we note that the ratio 
$-q_{i-1}/p_{i-1}$ represents the \emph{slope} of the 
line through $v_i$ in the coordinate system relative to the normalized
basis $\{e_1,e_2\}$ for $\sigma$.  We claim that these slopes form
a strictly decreasing sequence for $i \ge 0$, which will imply the statement 
in part b.  

It is an easy consequence of the recurrences \eqref{8.HJrecurr}
and the fact that $b_i \ge 2$ for all $i$ that the $p_i$ and 
$q_i$ are increasing sequences of nonnegative numbers.  You will show
this in Exercise~\ref{8.pqinc}.  

This applies to the slopes as
follows.   From \eqref{8.deteq},
\[
-q_i/p_i = -q_{i-1}/p_{i-1} - 1/(p_{i-1} p_i) < -q_{i-1}/p_{i-1},
\]
since $p_{i-1} p_i > 0$.  
Thus, $\sigma_i$ and $\sigma_{i+1}$ lie on opposite sides of 
$\Cone(v_i)$ for each $i$, and part b follows.

Part c follows from part b noting that $v_0 = e_2$ and 
$p_r/q_r = m/k$, so $v_{r+1} = m e_1 - ke_2$.
Hence the cones $\sigma_i$ fill out $\sigma$.

Part d follows from part c by the same reasoning used in the 
examples in \S 1 of this chapter.
\end{proof}

\begin{example}
\label{8.resex}
Consider the cone $\sigma = \Cone(e_2, 7e_1 - 5e_2)$ in normal
form.  To construct the resolution of the singularity of the
affine toric surface $U_\sigma$, we simply compute the Hirzebruch-Jung
continued fraction expansion of the rational number $m/k = 7/5$
using the modified Euclidean algorithm:
\begin{eqnarray*}
7 & = & 2\cdot 5 - 3\\
5 & = & 2\cdot 3 - 1\\
3 & = & 3\cdot 1.
\end{eqnarray*}
Hence $b_0 = b_1 = 2, b_2 = 3$, and
\begin{equation}
\label{8.sevenfive}
7/5 = [[2, 2, 3]].
\end{equation}
Then from Proposition~\ref{8.HJconvprop}  we have
\begin{eqnarray*}
p_0 = 1 & \qquad & q_0 = 0\\
p_1 = 2 & \qquad & q_1 = 1\\
p_2 = -p_0 + b_2p_1 = 3 & \qquad & q_2 = -q_0 + b_2q_1 = 2 \\
p_3 = -p_1 + b_3p_2 = 7 & \qquad & q_3 = -q_1 + b_3q_2 = 5.
\end{eqnarray*}
By Theorem~\ref{8.HJres}, the cones in the refinement $\Sigma$ of $\sigma$ 
giving the resolution of singularities are 
\begin{eqnarray}
\label{8.resexcones}
\sigma_1 & = & \Cone(e_2, e_1)\nonumber\\
\sigma_2 & = & \Cone(e_1, 2e_1 - e_2)\\
\sigma_3 & = & \Cone(2e_1 - e_2, 3e_1 - 2e_2)\nonumber\\
\sigma_4 & = & \Cone(3e_1 - 2e_2, 7e_1 - 5e_2)\nonumber.
\end{eqnarray}
We can also see an interesting connection between this Hirzebruch-Jung
continued fraction and the corresponding continued fraction expansion for $7/3$, 
where $5\cdot 3 \equiv 1 \bmod 7$ as in Proposition~\ref{8.cnfuniqueness}.
We leave it to the reader to check that 
\[
7/3 = [[3, 2, 2]],
\]
in which the partial quotients are the same as those in \eqref{8.sevenfive},
but listed in reverse order.   We will return to this point shortly.
\eoe
\end{example}

You will see several other explicit examples of this process in
Exercises~\ref{8.HJrncrdp} and \ref{8.wpp}.  
Our next result gives an alternative, more 
intrinsic, way to understand
the vectors $v_i$ in Theorem~\ref{8.HJres}.  Note that as is true for
$\sigma^\vee \cap M$, the set $\sigma \cap N$ has the structure of an 
additive semigroup.  We define its \emph{irreducible elements} and the 
\emph{Hilbert basis} as in Proposition~\ref{1.hilbbasis} from 
Chapter~\ref{affine}.  See Figure~2 in Example~\ref{8.rdpcont}
for an example.

\begin{theorem}
Let $\sigma = \Cone(e_2, m e_1 - k e_2)$ be in normal form.  Then
\[
S = \{v_0, v_1, \ldots, v_{r+1}\}
\] 
constructed in Theorem~\ref{8.HJres}
is precisely the Hilbert basis (the set of irreducible elements)
in the semigroup $\sigma \cap N$.  
\end{theorem}

\begin{proof}
Since each cone $\sigma_i$ is smooth,
$\sigma_i \cap N$ is generated by its ray generators
$\{v_{i-1},v_i\}$.  Since 
$\sigma = \sigma_1 \cup \cdots \cup \sigma_{r+1}$, it follows
that $S$ generates $\sigma \cap N$.  

We claim next that all the $v_i$ are irreducible elements
of $\sigma \cap N$.  This is clear for $v_0 = e_2$ and 
$v_{r+1} = me_1 - ke_2$ since they are the ray generators
for $\sigma$.   If $1\le i \le r$ and $v_i$ is not irreducible,  
then $v_i$ would have to be a linear combination of the vectors in 
$S \setminus \{v_i\}$ with nonnegative integer coefficients.
We will now use the result of Exercise~\ref{8.pqinc}, namely
that the sequences $p_i$ and $q_i$ are increasing sequences
of nonnegative numbers.  This means that if 
\[
v_i = p_i e_1 - q_i e_2 = \sum_{j\ne i} c_j v_j = 
\left(\sum_{j\ne i} c_j p_j\right) e_1 - \left(\sum_{j\ne i} c_j p_j\right)e_2
\]
with $c_j \ge 0$ in $\Z$, then 
\[
p_i = \sum_{j\ne i} c_j p_j, \qquad q_i = \sum_{j\ne i} c_j q_j.
\]
But the fact that $p_i$ and $q_i$ are increasing and $c_j \ge 0$ implies that
$c_j = 0$ for all $j > i$.  But this would imply that $v_i$ is a linear
combination with nonnegative integer coefficients of the vectors in 
$\{v_0,\ldots,v_{i-1}\}$.
This contradicts the fact from the proof of Theorem~\ref{8.HJres}
that the slopes of the $v_i$ are
strictly decreasing, and it follows that all the $v_i$ are irreducible elements
of the semigroup $\sigma \cap N$.  

Finally, we must show that there are no other irreducible elements
in $\sigma\cap N$.  But this follows from what we have already said.
Since $\sigma = \sigma_1 \cup \cdots \cup \sigma_{r+1}$, if $v$ is
any irreducible element, then $v \in \sigma_i \cap N$ for some $i$.
But then $v = c_{i-1} v_{i-1} + c_i v_i$ for some $c_{i-1},c_i \ge 0$
in $\Z$.  Thus $v$ is irreducible only if $c_{i-1} = 1$ and $c_i = 0$
or vice versa.  
\end{proof}

Next, we will show how the irreducible elements $v_i$ in $\sigma \cap N$
determine the partial quotients in the Hirzebruch-Jung continued
fraction expansion of $m/k$.  

\begin{theorem} 
\label{8.selfinter}
Let $\sigma = \Cone(m e_1 - ke_2)$ be in normal form, 
and let 
\[
m/k = [[b_1, b_2, \ldots, b_r]].  
\]
The vectors
$\{v_0, v_1, \ldots, v_{r+1}\}$ constructed in Theorem~\ref{8.HJres}
satisfy
\begin{equation}
\label{8.intereq}
v_{i-1} + v_{i+1} = b_i v_i
\end{equation}
for all $i$, $1 \le i \le r$.  
\end{theorem}

\begin{proof}
By the recurrences \eqref{8.HJrecurr}, 
\begin{eqnarray*}
v_{i-1} + v_{i+1} & = & (p_{i-2} e_1 - q_{i-2} e_2)  + (p_{i} e_1 - q_{i}e_2) \\
& = & (p_{i-2} + p_{i})e_1 - (q_{i-2} + q_{i})e_2\\
& = & b_{i} (p_{i-1} e_1 - q_{i-1} e_2)\\
& = & b_{i} v_{i}, 
\end{eqnarray*}
which is what we wanted to show.  
\end{proof}

In the next section we will see several important consequences
of the equations \eqref{8.intereq}  connected with the geometry of the smooth toric
surface $X_\Sigma$.  To conclude this section, we will show that
the pattern noted at the end of Example~\ref{8.resex} is a general phenomenon 
about Hirzebruch-Jung continued fractions, and this can be proved in a nice way 
from the properties of the associated toric surfaces we have developed.

\begin{proposition}
\label{8.reverse}
Let $0 < k, \widetilde{k} < m$ and assume $k\widetilde{k} \equiv 1 \bmod m$.
If the Hirzebruch-Jung continued fraction expansion of $m/k$ is 
\[
m/k = [[b_1, b_2, \ldots, b_r]],
\]
then the Hirzebruch-Jung continued fraction expansion of $m/\,\widetilde{k}$ is
\[
m/\,\widetilde{k} = [[b_r, b_{r-1}, \ldots, b_1]].
\]
\end{proposition}

\begin{proof}
Let $\sigma = \Cone(e_2, me_1 - ke_2)$ and 
$\widetilde{\sigma} = \Cone(e_2, me_1 - \widetilde{k}e_2)$
be the corresponding cones in normal form.   Since $k\widetilde{k} \equiv 1 \bmod m$, 
there is an integer $\widetilde{m}$ such that $m\widetilde{m} + k\widetilde{k} = 1$.
The $\Z$-linear mapping $\varphi: N \to N$ defined with respect to the basis
$\{e_1,e_2\}$ by the matrix 
\[
A = \begin{pmatrix} \widetilde{k} & m\\
                    \widetilde{m} & -k
    \end{pmatrix}
\]
is bijective, maps $\widetilde{\sigma}$ to $\sigma$, and is orientation-reversing
so $\varphi(me_1 - \widetilde{k} e_2) = e_2$ and $\varphi(e_2) = me_1 - ke_2$.
If we apply Theorem~\ref{8.HJres} to $\sigma$, then we obtain vectors 
$v_i$ satisfying the equations 
\[
v_{i-1} + v_{i+1} = b_i v_i
\]
for all $1 \le i \le r$.  We claim that when we apply the mapping $\varphi^{-1}$
defined by the inverse of the matrix $A$ above, then the vectors $v_i$ 
are taken to corresponding vectors $\widetilde{v}_i$ for the cone $\widetilde{\sigma}$.
But since $\varphi$ and $\varphi^{-1}$ are orientation-reversing, the partial
quotients in the Hirzebruch-Jung continued fraction will be listed
in the opposite order.  You will complete the proof of this assertion
in Exercise~\ref{8.revord}.
\end{proof}

\exercises

\small
\begin{exercise}
\label{8.ordconvex}  
In this problem we will consider the statement in 
Proposition~\ref{8.cfracconvprop}.
\begin{thmenum}
\item Show that the sequences $p_i$ and $q_i$ satisfy
\[
p_{i-1}q_i - p_iq_{i-1} = -1
\]
for all $i$.  Hint: see the proof of part a of Theorem~\ref{8.HJres}.
\item Prove the proposition. 
Hint:  Argue by induction on the length $n\ge 1$ of the expansion.
The formal expression $[a_1, a_2, \ldots, a_i]$ is well-defined even
if the $a_l$ are not integers, so we can write
\[
[a_1, a_2, \ldots, a_{i-1}, a_i] = [a_1, a_2, \ldots, a_{i-1} + 1/a_i].
\]
Then use the recurrences in \eqref{8.ordcfracrecurr}.  For 
the statement about the $\gcd(p_i,q_i)$, use part a.
\item Suppose we modify the initialization and the recurrences
\eqref{8.ordcfracrecurr} as follows.  Let
\begin{eqnarray*}
r_{-1} = 1 & \qquad & s_{-1} = 0\\
r_0 = 0 & \qquad & s_0 = 1.
\end{eqnarray*} 
Then, for all $1\le i \le n$, compute
\begin{eqnarray*}
r_i & = & r_{i-2} - a_i r_{i-1}\\
s_i & = & s_{i-2} - a_i s_{i-1}
\end{eqnarray*}
(note the change in sign!)
What is true about $r_i m + s_i k$ for all $i$?  What do we
get with $i = n$?  Hint:  This fact is the basis for the 
Extended Euclidean Algorithm.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{8.arbcfrac}  
In this exercise, we will study the analogs
of the continued fraction expansions from \eqref{8.cfrac} for 
arbitrary reals.  If $\alpha$ is a real number, then we say
\[
\alpha = [a_1, a_2, a_3, \ldots ]
\]
if, by analogy with \eqref{8.cfraccomps}, $\alpha = a_1 + \alpha_1$,
where $a_1$ is the greatest integer less than or equal to $\alpha$
and $0 \le \alpha_1 < 1$.  Then for all $i \ge 1$, 
\[
1/\alpha_i  =  a_{i+1} + \alpha_{i+1},
\]
where at each step $a_i > 0$ is an integer, and $0 \le \alpha_i < 1$ 
is the fractional part of $1/\alpha_{i-1}$.   Most computer algebra 
systems can compute continued fraction expansions.  
\begin{thmenum}
\item Show that the result of Proposition~\ref{8.cfracconvprop} holds
even when the continued fraction expansion does not terminate.  
\item Compute the continued fraction expansions of $\alpha = \sqrt{2}$ and 
$\alpha = \sqrt{3}$.  Do you notice something interesting?
\item Show that if the continued fraction expansion of $\alpha$ is 
eventually periodic, then $\alpha$ satisfies an irreducible
quadratic polynomial with rational coefficients.  The converse of this
statement is also true, but somewhat more difficult to prove. 
\item Compute $a_0, \ldots, a_{20}$ for $\alpha = e$.  Do you notice
something interesting? 
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{8.convert}
A key ingredient for a method for converting ordinary
continued fraction expansions to Hirzebruch-Jung continued
fractions is the following observation.  
\begin{thmenum}
\item Given $a_0, a_1 > 0$ 
and a variable $x$, show that as rational expressions
\[
[a_1, a_2, x] = [[a_1 + 1, (2)^{a_2-1}, x + 1]],
\]
where for any $l\ge 0$, $(2)^l$ denotes a string of $l$ $2$'s.
Hint: argue by induction on $a_1$.
\item Show how this observation yields the equality
\[
[1, 1, 1, 5] = [[2, 3, 2, 2, 2, 2]]
\]
from Example~\ref{8.cfcompare}.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{8.HJconvex}
Prove Proposition~\ref{8.HJconvprop} by adapting the proof from 
Exercise~\ref{8.ordconvex}.
\end{exercise}

\begin{exercise}
\label{8.pqinc}
Show that the sequences $p_i$ and $q_i$ from \eqref{8.HJrecurr}
are increasing sequences of nonnegative numbers.  Hint: recall
that $b_i \ge 2$ for all $i$.   
\end{exercise}

\begin{exercise}
\label{8.intereqex}
Verify that equations \eqref{8.intereq} hold for
the vectors $v_i$ computed in Example~\ref{8.resex}.  (See 
\eqref{8.resexcones}.)
\end{exercise}

\begin{exercise}
\label{8.revord}
Verify the last claim in the proof of Proposition~\ref{8.reverse}.
\end{exercise}

\begin{exercise}
\label{8.HJrncrdp}
In \S 1, we constructed several resolutions of singularities
in a rather \emph{ad hoc} way.  In this exercise, we will see
that the resolutions given by Theorem~\ref{8.HJres} are the same as what
we saw before.
\begin{thmenum}
\item Show that the Hirzebruch-Jung continued fraction method
gives exactly the same resolution of the singular point $p_\sigma$
for $\sigma$ with parameters $m, 1$ as that given in Example~\ref{8.rnccont}.
\item Do the same for the resolutions of the singular points for 
$\sigma$ with parameters $m, k = m-1$ (the rational double points
from Example~\ref{8.rdpcont}).  Hint: You will want to start by 
showing that
\[
m/(m-1) = [[2,2,\ldots,2]],
\]
where there are $m - 1$ $2$'s.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{8.wpp}
In this exercise, you will apply the results of this section
to the weighted projective planes $\P(q_0,q_1,q_2)$ from 
Chapter~\ref{poly} and Example~\ref{3.weightedproj} in Chapter~\ref{normal}.
\begin{thmenum}
\item Construct a resolution of singularities for any
$\P(1,1,q_2)$, where $q_2 \ge 2$.  What smooth toric surface
is obtained in this way?  A complete classification of the 
smooth complete toric surfaces will be developed in \S 3 of this
chapter.
\item Do the same for $\P(1,q_1,q_2)$ in general.  
\end{thmenum}
\end{exercise}

\normalsize
\section{Smooth Toric Surfaces}

\subsection*{\emph{\bfseries Classification of Smooth Toric Surfaces}} We
will begin this section by using the results of \S\S 1 and 2 to give
a complete classification of the smooth complete toric surfaces.  Our
theorem will say that these are all obtained by toric blowups
from either $\P^2, \P^1\times \P^1$,
or one of the Hirzebruch surfaces $\cH_r$ with $r \ge 2$ from 
Example~\ref{3.hirzebruch} in Chapter~\ref{normal}.  The proof
will be based on the following facts.

First, it follows from Proposition~\ref{3.starblowup} of Chapter~\ref{normal}
that if $\sigma = \Cone(v_1,v_2)$ is a smooth cone and we refine
$\sigma$ by inserting the new one-dimensional cone $\tau = \Cone(v_1+v_2)$,
then on the resulting toric surface, the smooth point $p_\sigma$ will be
blown up to a copy of $\P^1$. 

For the second ingredient of the proof, we introduce the following
notation.  If $\Sigma$ is a smooth complete fan, 
then list the ray generators of the two-dimensional cones 
in $\Sigma$ as $v_0, v_1, \ldots, v_{d-1}$ in clockwise order around 
the origin in $N_\R$, and we will consider the indices as integers modulo 
$d$, so $v_d = v_0$.  Then we have the following statement parallel to \eqref{8.intereq}.

\begin{lemma}
\label{8.raygens} 
Let $v_0,\ldots,v_d$ be the ray generators
for a smooth complete fan $\Sigma$ in $N_\R \simeq \R^2$.
For all $i$, there exist integers $b_i$, $i = 0, \ldots, d-1$ such
that 
\begin{equation}
\label{8.lincomb}
v_{i-1} + v_{i+1} = b_i v_i.
\end{equation}
\end{lemma}

\begin{proof}
All the cones in $\Sigma$ are assumed to be smooth.  Hence $\{v_{i-1},v_i\}$
is one $\Z$-basis for $N$ and we can write
\[
v_{i+1} = a_i v_{i-1} + b_i v_i
\] 
for some integers $a_i,b_i$.  By our choice of indices
$\{v_i,v_{i+1}\}$ is another ordered $\Z$-basis for $N$
with the same orientation as $\{v_{i-1},v_i\}$.  
Hence
\[
\det \begin{pmatrix} 0 & a_i \\
                     1 & b_i 
     \end{pmatrix} 
= +1.
\]
It follows that $a_i = -1$ and $v_{i-1} + v_i = b_i v_i$ as claimed.
\end{proof}

A consequence of the proof of this fact and the results of \S 2 is 
the following statement.

\begin{lemma}
\label{8.smoothref}
Let $\Sigma$ be a smooth fan which is a refinement of a smooth cone
$\sigma$.  Then $\Sigma$ is obtained from $\sigma$ by a sequence
of star subdivisions as in Definition~\ref{3.starsubdivision}
from Chapter~\ref{normal}.
\end{lemma}

\begin{proof}
Say there are $r$ two-dimensional cones in $\Sigma$
and the ray generators in are $v_0,v_1,\ldots,v_r$, listed
clockwise starting from $v_0$.  We argue by induction on $r$.
If $r = 1$, there is only one cone in $\Sigma$ and there is 
nothing to prove.  Assume the result has been proved for all $\Sigma$
with $r = s$ two-dimensional cones, and consider the case $r = s+1$.
The proof of Lemma~\ref{8.raygens}
works locally in the fan, so under the hypotheses here, for all $i$,
$1 \le i \le s$, 
there exist integers $b_i$ such that \eqref{8.lincomb} holds.
Note that $\sigma$ is strongly convex so $b_i > 0$ for all $i$.
We claim that there exists some $i$ such that $b_i = 1$.  
If not, that is, if $b_i \ge 2$ for all $i$, then as in \S 2, 
the Hirzebruch-Jung continued
fraction 
\[
[[b_1, b_2, \ldots, b_s]]
\]
represents a rational number $m/k$ and the cone $\sigma$ has parameters
$m,k$ with $m > k > 0$.  But then $m \ge 2$, which would contradict the assumption
that $\sigma$ is a smooth cone.  

Hence there exists an $i$, $1 \le i \le s$, such that 
\[
v_{i-1} + v_{i+1} = v_i.
\]
In this situation, $\Cone(v_{i-1},v_{i+1})$ is also smooth. 
You will verify this claim in Exercise~\ref{8.smoothrefex}.
Moreover, $\Cone(v_{i-1},v_i)$ and $\Cone(v_i,v_{i+1})$ are precisely
the cones in the star subdivision of  $\Cone(v_{i-1},v_{i+1})$.  
Hence the proof is complete by induction.
\end{proof}

We are now ready to state our classification theorem.

\begin{theorem}
\label{8.smoothclass}
Every smooth complete toric surface $X_\Sigma$ is obtained
from either 
\[
\P^2, \P^1 \times \P^1,\ \mathrm{or}\ \cH_r, r \ge 2
\] 
by a finite sequence of blowups at fixed points of
the torus action.  
\end{theorem}

\begin{proof}
We label the ray generators of the fan $\Sigma$ as
in Lemma~\ref{8.raygens}.  As in the proof
of Lemma~\ref{8.smoothref}, if $b_i = 1$ in 
\eqref{8.lincomb} for some $i$, then our
surface is a blowup of the surface corresponding
to the fan where $v_i$ is removed.  Hence 
we only need to consider the case 
in \eqref{8.lincomb} where $b_i \ne 1$ for all $i$.  

\begin{figure}[ht]
\begin{center}
\includegraphics[height=1.7truein]{Chapter8Fig3.ps}
\end{center}
\caption{The ray generators $v_0,v_1,v_2,v_j$ when $v_j = -v_1$} 
\end{figure}
Suppose first that $v_j = -v_i$ for some $i < j$.  We
relabel the vertices to make $v_j = -v_1$ for some $j$.
Note that $j > 2$ since the cones must be strongly convex.  
Then from \eqref{8.lincomb},
\[
v_0 = -v_2 + b_1 v_1.
\]
These ray generators are shown in Figure 3, relative to
the $\Z$-basis $\{v_1,v_2\}$ for $N$.  Comparing 
this with the fans for $\cH_r$ and $\P^1 \times \P^1$
(Figures 3 and 4 from Chapter~\ref{normal}, \S 1),
we see that $\Sigma$ must be a refinement of the 
fan for $\cH_r$ if $r = b_1 > 2$.  The
same follows if $b_1 < -2$ and $r = |b_1|$ (see
Exercise~\ref{8.nega1}).  If $b_1 = 0$, then
we have a refinement of the fan for
$\P^1 \times \P^1$.  Hence the theorem follows from 
Lemma~\ref{8.smoothref} in the
case $v_j = -v_1$ for some $j$.


Now suppose $v_j \ne -v_i$ for all $j > i$.  Consider 
the $\Z$-basis $\{v_1,v_2\}$ for $N$ and the possible
locations of $v_3$. 
We will use the usual terminology for the quadrants 
in $N_\R$ relative to the axes of the real coordinate
system with the real span of $v_1$ playing the role of the 
$y$-axis and the real span of $v_2$ playing the role of the
$x$-axis.  However, the ordered basis $\{v_1,v_2\}$ has the 
opposite of the usual orientation.
This means, for instance,
that the open fourth quadrant will consist of vectors 
\[
w = a v_1 + b v_2
\]
with $a < 0$ and $b > 0$.
 Since $\Cone(v_2,v_3)$ must
be strongly convex, $v_3$ must lie in the interior
of either the third or fourth quadrant of the plane
If $v_3$ is in the \emph{third} quadrant, then
from \eqref{8.lincomb},
\[
v_3 = -v_1 + b_2 v_2,
\]
where $b_2 < 0$.  

If $b_2 = -1$, then $\Cone(v_3,v_1)$
is smooth.  It follows from our simplifying assumptions
based on Lemma~\ref{8.smoothref} that $v_0 = v_3$ and
we see from Figure 2 of Chapter~\ref{affine} that 
$\Sigma$ must be the standard fan for $\P^2$.  (If 
we did not invoke Lemma~\ref{8.smoothref}, the
conclusion would be that $\Sigma$ was a refinement
of the fan for $\P^2$.)  The theorem follows in this case.  

We claim that the case $b_2 < -1$ cannot occur
under the assumption $v_j \ne -v_j$.  We may use the $\Z$-basis 
$f_1 = -v_2, f_2 = v_1$ for $N$.  Writing
$m = -b_2$, we have 
\[
\Cone(v_3,v_1) = \Cone(mf_1 - f_2, f_2),
\]
which is in normal form with parameters $m, 1$, but
with the reverse of the usual orientation.   The
other vectors $v_4, \ldots, v_d = v_0$ come from
a smooth refinement of this cone.  Hence by 
Proposition~\ref{8.reverse}, we must have 
\[
m = m/1 = [[b_d, b_{d-1},\ldots,b_4]].
\]
However, by the modified
division algorithm from \S 2, 
this implies that $d = 4$ and $b_4 = -b_2 = m$.
We have 
\[
-b_2v_0 = b_4v_0 = b_4v_4 = v_3 + v_5 = v_3 + v_1 = b_2 v_2.
\]
This would imply $v_2 = -v_0$, which is a contradiction.

Finally, if $v_3$ is in the \emph{fourth} quadrant, consider
the last ray generator $v_j$, $j \ge 3$ in the fourth
quadrant, counting clockwise from $v_2$.  We have
\begin{equation}
\label{8.lastin4}
v_j = a v_2 - b v_1,
\end{equation}
where $a,b > 0$.  The next vector $v_{j+1}$ must lie
to the left of the line containing $v_1$ (since we are
assuming $v_{j+1} \ne -v_1$).  Hence 
\begin{equation}
\label{8.next}
v_{j+1} = -c v_2 + d v_1,
\end{equation}
where $c > 0$.  The case $d = 0$ cannot occur under our
assumption $v_j \ne -v_i$.   We claim that $d \ne 0$
also leads to a contradiction.  Note that by the hypotheses
$\Cone(v_j, v_{j+1})$ is smooth and has the same orientation
as $\Cone(v_1,v_2)$.  Hence the constants $a,b,c,d$
from \eqref{8.lastin4} and \eqref{8.next} satisfy
\[
\det \left(\begin{array}{rr} -b & d\\
                              a & -c \end{array}\right) = bc - ad = 1.
\]
But $a,b,c,d$ are all integers and $a,b,c > 0$.  If $d < 0$ then the
determinant is strictly larger than 1.  If $d > 0$,
then $v_{j+1}$ lies in the \emph{second} quadrant and
\[
v_i + v_{j+1} = (a-c)v_2 + (d-b)v_1,
\]
lies in the \emph{third} quadrant, so
$a - c < 0$ and $d - b < 0$.  But then $c > a > 0$ and $b > d > 0$.  
Hence $bc - ad$ must be strictly larger than 1 again.
You will check the details of the last part of this argument
in Exercise~\ref{8.thmex}.
\end{proof}

Since there is also a Hirzebruch surface $\cH_1$, the statement
of this theorem might seem puzzling.  The reason that $\cH_1$
is not included is that this surface is actually a 
blowup of $\P^2$.  See Exercise~\ref{8.F1}.

The problem of classifying all higher-dimensional toric varieties
is much more difficult. No complete classifications are known,
although results for fans with small numbers of one-dimensional
cones relative to the dimension of $N_\R$ have been developed in 
\cite{batyrev}.


\subsection*{\emph{\bfseries Divisors on a Smooth Toric Surface}}
Our next results will deal with the geometry of divisors on
smooth complete toric surfaces.  We begin by summarizing how the general 
facts on divisors and line bundles on toric varieties
from Chapters~\ref{divisors} and \ref{invertible} specialize
to the case of a smooth complete toric surface $X_\Sigma$.  (**add references
later**)

\begin{coxitem}
\item First, since $X_\Sigma$ is smooth, Weil and Cartier divisors 
coincide.  
\item The group $\Pic(X_\Sigma)$ of line bundles modulo isomorphism
coincides with the group of divisor classes $\Cl(X_\Sigma)$, and 
there is an exact sequence
\begin{equation}
\label{8.picseq}
0 \longrightarrow M \longrightarrow \oplus_{i=1}^d \Z\cdot D_i \longrightarrow \Pic(X_\Sigma) \longrightarrow 0,
\end{equation}
where the $D_i$ are the divisors corresponding to the one-dimensional
cones $\Cone(v_i)$ in $\Sigma$ and $d$ is the number of edges.  The 
mapping to $\oplus_{i=1}^d \Z\cdot D_i$ takes $u \in M$ to 
\[
\mathrm{div}(\chi^u) = \sum_{i=1}^d \langle u,v_i\rangle D_i.
\]
The mapping
to $\Pic(X_\Sigma)$ 
is the map sending $D$ to the isomorphism class of the line bundle
$\cO(D)$.  
\item From this exact sequence we have an isomorphism 
\[
\Pic(X_\Sigma) \simeq \Z^{d-2}.
\]
\item The canonical divisor class on $X_\Sigma$ is the class of
\[
K = -(D_1 + \cdots + D_d),
\] 
where as before the $D_i$
are the divisors corresponding to the one-dimensional cones in $\Sigma$.  
\end{coxitem}

\begin{example}
\label{8.HirzPic}
For the Hirzebruch surface $\cH_r$, use the
standard fan as in Figure~4 from Chapter~\ref{normal}, and let
$D_1$ correspond to $\Cone(e_2)$, $D_2$ correspond to $\Cone(e_1)$,
$D_3$ correspond to $\Cone(-e_2)$ and $D_4$ correspond to $\Cone(-e_1 + re_2)$.
Then 
\[
\Pic(\cH_r) \simeq \oplus_{i=1}^4 \Z\cdot D_i / (\Z\cdot(D_2 - D_4) \oplus \Z\cdot(D_1 - D_3 + r D_4)).
\]
Hence $D_4 \equiv D_2$ and $D_3 \equiv D_1 + r D_2$, so 
it follows that $\Pic(\cH_r) \simeq \Z^2$ is freely generated by the classes of $D_1$ and $D_2$.  
Moreover, the canonical divisor is
\[
K = -(D_1 + D_2 + D_3 + D_4) \equiv -2 D_1 - (r+2) D_2.
\] 
\eoe
\end{example}

\subsection*{\emph{\bfseries The Intersection Product}}
A fundamental feature of the theory of smooth algebraic surfaces is
the intersection product on divisors or divisor classes.  Let
$C$ and $D$ be two curves on a smooth surface $S$ having no irreducible
components in common, so $C \cap D$ is a finite
collection of points.  At each $p\in C\cap D$, we define a
local intersection number using local equations $f_p = 0$ for $C$
and $g_p = 0$ for $D$ at $p$: 
\begin{equation}
\label{8.locint}
(C\cdot D)_p = \dim_\C \cO_{S,p}/\langle f_p, g_p \rangle. 
\end{equation}
Then the global intersection number of $C$ and $D$ is the sum
of the local intersection numbers:
\begin{equation}
(C\cdot D) = \sum_{p \in C\cap D} (C\cdot D)_p.
\label{8.globint}
\end{equation}

\begin{example}
\label{8.rayinters}
For instance, say $\sigma$ is any smooth cone in a complete 
smooth fan $\Sigma$.  Let  $\rho$ and $\tau$ be the edges of $\sigma$,
and let $C = \overline{O(\rho)}$, $D = \overline{O(\tau)}$ 
be the corresponding divisors on $X_\Sigma$.  Then $C \cap D = \{p_\sigma\}$,
the $T_N$-fixed point corresponding to $\sigma$.  Moreover
by a change of basis in $N$, we can assume $\rho = \Cone(e_1)$ and
$\tau = \Cone(e_2)$, so $C, D$ are defined in this
coordinate system by the equations $x = 0$ and $y = 0$.
Then \eqref{8.locint} shows
\[
(C \cdot D)_{p_\sigma} = \dim_\C \C[x,y]_{(0,0)} / \langle x,y\rangle = 1.
\]
and hence $(C\cdot D) = 1$.  

On the other hand, if $\rho$ and $\tau$ are not edges of the 
same two-dimensional cone, then in Exercise~\ref{8.nointer}
you will show that $(C\cdot D) = 0$.
\eoe
\end{example}

Our goal is to extend this intersection product to a $\Z$-valued bilinear
form on the group of divisor classes on $X_\Sigma$, or $\Pic(X_\Sigma)$.
The obstacle is that \eqref{8.globint} does not apply when $C = D$, since
then the set-theoretic intersection $C \cap D$ is not a finite set of
points.  To define $(C\cdot C)$, we must proceed in a different way.
This topic is discussed in detail for general surfaces
in \cite[Chapter V]{hartshorne} and in \cite[Chapter VI]{shafarevich}.

For any curve $C$ in a smooth surface $S$, we have the sheaf of 
ideals defining $C$,
\[
\cI_C \simeq \cO(-C).
\] 
(This isomorphism comes from the
fact that $C$ defines a Cartier divisor.)  The inclusion $i : C \hookrightarrow S$
gives an exact sequence of sheaves
\[
0 \longrightarrow \cI_C \longrightarrow \cO_S \longrightarrow  i_*\cO_C \longrightarrow 0,
\]
where the sections of the sheaf $i_*\cO_C$ over $U \subseteq S$ are 
$i_*\cO_C(U) = \cO_C(C\cap U)$.  The sheaf $\cI_C/\cI_C^2$ is called the 
\emph{conormal sheaf} of $C$ in $S$ and its dual
\[
\cN_{C/S} = \underline{Hom}_{\cO_S} (\cI_C/\cI_C^2, \cO_S)
\]
is called the \emph{normal sheaf} of $C$ in $S$.  In this situation,
$\cN_{C/S}$ is an invertible sheaf on $C$.  Its sections
can be interpreted as giving first-order deformations of $C$ in $S$,
so intuitively the degree of $\cN_{C/S}$ on $C$ measures the 
intersections of $C$ and a ``nearby'' curve equivalent to $C$.      
This is the reason $\cN_{C/S}$ will be used to define the 
intersection number $(C \cdot C)$.

In our situation on a smooth complete toric surface
this can all be made very concrete.  First, for each
one-dimensional cone $\rho$, the 
divisor $D_\rho$ satisfies 
\[
D_\rho \simeq \P^1
\] 
as a curve, and 
we have the standard degree isomorphism 
\[
\deg: \Pic(\P^1) \simeq \Z
\]
from (**ref**).   

\begin{definition} 
\label{8.selfinterdef} 
 Let $D_\rho$ be a divisor as above on the smooth 
toric surface $X_\Sigma$.
The \emph{\bfseries self-intersection number} 
$(D_\rho\cdot D_\rho)$ is defined as
\[
(D_\rho\cdot D_\rho) = \deg(\cN_{D_\rho/X_\Sigma}).
\]
\end{definition}

Moreover, these self-intersection numbers are determined
by the fan $\Sigma$ in the following way.

\begin{theorem}  
\label{8.selfintno}
Let $D = D_\rho$ be the divisor on 
a smooth complete toric surface corresponding to 
$\rho = \Cone(v)$ which is a face
of the smooth two-dimensional cones $\sigma_1 = \Cone(u,v)$ and 
$\sigma_2 = \Cone(v,w)$.  Then
\[
(D \cdot D) = -b,
\]
where $u + w = bv$ as in \eqref{8.lincomb}.
\end{theorem}

\begin{proof}
We use $e_1 = w$ and $e_2 = v$ as our $\Z$-basis for 
the lattice $N$.  Then from \eqref{8.lincomb},
\[
u = a v - w = -e_1 + b e_2.
\]
Write $x,y$ for the affine coordinates in $U_{\sigma_2}$.  Then
$D$ is defined in $U_{\sigma_2}$ by the equation $y = 0$.  Similarly,
$D$ is defined in $U_{\sigma_1}$ by the equation $x^b y = 0$.
Thus the stalk of $\cI_D/\cI_D^2$ over each point in $U_{\sigma_2} \cap D$ is 
one-dimensional and generated by $y$.  Similarly $x^b y$ generates
the stalk of $\cI_D/\cI_D^2$ over each point 
$U_{\sigma_1} \cap D$.  Since $x$ is the coordinate
on $D$, this means that under the isomorphism $D \simeq \P^1$, 
\[
\cI_D/\cI_D^2 \simeq \cO_{\P^1}(b),
\]
which has degree $b$.  The normal sheaf $\cN_{D/X_\Sigma}$ is
the dual of this, so 
\[
\deg(\cN_{D/X_\Sigma}) = -b,
\]
which is what we wanted to show.  
\end{proof}

\begin{example}
\label{8.excfirst}
Let $\sigma = \Cone(v_1,v_2)$ be a smooth cone in a fan $\Sigma'$, 
and consider the star subdivision, in which $\Cone(v_1+v_2)$ is 
inserted to subdivide $\sigma$ into two cones.  Call the refined
fan $\Sigma$.  The exceptional divisor $E$ of the blowup
\[
\pi : X_\Sigma \to X_{\Sigma'}
\]
satisfies $(E\cdot E) = -1$ on $X_\Sigma$.  Curves with 
self-intersection number $-1$ on a surface are called
\emph{exceptional curves of the first kind}.  They
can always be \emph{contracted}
to a smooth point on a birationally equivalent surface, as in 
this toric example.  
\end{example}

One of the foundational results in the theory of general 
algebraic surfaces is that every function field of transcendence 
degree 2 has a \emph{relatively minimal model}, that
is, a surface $S$ such that if $\pi : S \to S'$ is 
a birational morphism, then $\pi$ is necessarily an
isomorphism.  The proof is given in \cite[V.5.8]{hartshorne}.
Interestingly, the possible relatively minimal models for 
\emph{rational} surfaces (ones whose function field is isomorphic
to $\C(x,y)$) can be described in toric terms -- they are
precisely the surfaces $\P^2$, $\P^1\times \P^1$, and $\cH_r$
for $r\ge 2$ from Theorem~\ref{8.smoothclass}, and the proof 
of that theorem essentially shows that after contracting
all possible exceptional curves of the first kind, 
every smooth toric surface will be taken to one of these. 

\begin{example}
\label{8.rncexdiv}
Consider a complete singular toric surface with a singular
point $p_\sigma$ corresponding to a cone $\sigma$ with parameters
$m, 1$ for some $m > 1$.  If the singularity is resolved
as in Example~\ref{8.rnccont}, consider the exceptional
divisor $E$.  Using the normalized basis of $N$ relative 
to $\sigma$, the one-dimensional cone that is inserted 
in the refined fan is $e_1$, and 
\[
e_2 + (me_1 - e_2) = m e_1.
\]
Hence
\[
(E\cdot E) = -m
\]
gives the self-intersection number of $E$.  
\eoe
\end{example}

The intersection product can now be extended by linearity to 
all of $\Pic(X_\Sigma)$, and this yields a $\Z$-valued 
symmetric bilinear form.

\begin{example}
\label{8.Hirzform}
Consider the Hirzebruch surface $\cH_r$.  We follow the 
notation from Example~\ref{8.HirzPic} above.
We know that the group $\Pic(\cH_r)$ is generated by the classes
of $D_1$ and $D_2$.  By Example~\ref{8.rayinters} and 
Theorem~\ref{8.selfintno}, we have 
\[
(D_1 \cdot D_1) = -r, \quad (D_2 \cdot D_2) = 0, \quad (D_1 \cdot D_2) = (D_2 \cdot D_1) = 1.
\]
The intersection product is described
by the matrix
\[
\begin{pmatrix} (D_1\cdot D_1) & (D_1 \cdot D_2)\\
                (D_2\cdot D_1) & (D_2 \cdot D_2) 
\end{pmatrix} = \begin{pmatrix} -r & 1\\
                                 1 & 0
                \end{pmatrix}.
\]
If $D \equiv m D_1 + n D_2$ and $E \equiv pD_1 + qD_2$ are 
any two divisors on the surface, then 
\begin{equation}
\label{8.Hirzinterprod}
(D\cdot E) = \begin{pmatrix} m & n\end{pmatrix}
 \begin{pmatrix} -r & 1\\
                  1 & 0
                \end{pmatrix}
\begin{pmatrix} p \\ q \end{pmatrix} = -rmp + np + mq.
\end{equation}
For instance, with $D = E = D_3 = D_1 + rD_2$, we obtain
\[
(D_3\cdot D_3) = -r + r + r = r.
\]
The self-intersection number
$(D_2\cdot D_2) = 0$ is a reflection of the fibration structure on $\cH_r$
studied in Example~\ref{3.HP1Ex3} from Chapter~\ref{normal}.  The divisors
$D_2$ and $D_4$ are fibers of the mapping $\cH_r \to \P^1$.  Such curves always
have self-intersection equal to zero.
You will compute several other intersection products 
on $\cH_r$ in Exercise~\ref{8.Hirzex}.
\eoe
\end{example}

Now suppose the complete smooth toric surface $X_\Sigma$
is obtained via a resolution of singularities of some normal
toric surface containing a singularity $p_\sigma$ corresponding
to a two-dimensional cone $\sigma$ with parameters $m,k$.  
From Theorem~\ref{8.selfinter} and Theorem~\ref{8.selfintno}, 
if the Hirzebruch-Jung continued fraction expansion of $m/k$ is
\[
m/k = [[b_1, b_2, \ldots, b_r]],
\]
then the components of the exceptional divisor mapping to $p_\sigma$,
which will be denoted $D_1, \ldots, D_r$, satisfy
\begin{equation}
\label{8.excinter}
(D_i \cdot D_j) = \begin{cases} 
                  -b_i & \hbox{if}\quad j = i\\
                  1    & \hbox{if}\quad |i-j| = 1\\
                  0    & \hbox{otherwise}.
                  \end{cases}
\end{equation}
You will show in Exercise~\ref{8.negdef} that the matrix of the 
intersection product restricted to the subgroup of $\Pic(X_\Sigma)$
generated by the classes of the $D_i$ defines a \emph{negative definite} 
quadratic form.   This condition is necessary for \emph{contractibility} 
of a curve $C$ on a smooth surface $S$, that is, existence of a birational 
morphism $\pi : S \to X$, where $\pi(C)$ is a (possibly singular) point on 
$X$.  

\subsection*{\emph{\bfseries Rational Double Points Reconsidered}}
If $\sigma$ has parameters $m, m - 1$, then from Exercise~\ref{8.HJrncrdp},
the Hirzebruch-Jung continued fraction expansion of $m/(m-1)$ is
\[
m/(m-1) = [[2,2,\ldots,2]],
\]
with $m - 1$ terms.  Hence $b_i = 2$ for all $i$, and 
\eqref{8.excinter} gives the tridiagonal $(m-1) \times (m-1)$ matrix
\[
\left(\begin{array}{rrrrr}
-2 &  1 & 0 &  \cdots & 0\\
 1 & -2 & 1 &  \cdots & 0\\
 0 & \ddots & \ddots & \ddots & 0\\
 0 &      & 1 & -2 & 1\\
 0 &  \cdots  & 0 & 1 & -2 
\end{array}\right).
\]
representing the intersection product on the subgroup of $\Pic(X_\Sigma)$
generated by the components of the exceptional divisor for the 
resolution of a rational double point of type $A_{m-1}$.  At this
point we can fully explain the terminology for these singularities.

The problem of classifying lattices 
\[
\Z e_1 \oplus \cdots \oplus \Z e_s
\]
with negative definite bilinear forms $B$ satisfying 
$B(e_i,e_i) = -2$ for all $i$ arises in many areas within mathematics, 
most notably in the classification of complex simple Lie algebras 
via root systems.  The matrix above is the (negative of) the 
Cartan matrix for the root system type $A_{m-1}$, which is 
often represented by the Dynkin diagram:
\[
\begin{picture}(220,25)
\thicklines
\put(25,15){\circle*{5}}
\put(50,15){\circle*{5}}
\put(75,15){\circle*{5}}
\put(185,15){\circle*{5}}
\put(160,15){\circle*{5}}
\put(25,15){\line(1,0){70}}
\put(105,15){\line(1,0){5}}
\put(115,15){\line(1,0){5}}
\put(125,15){\line(1,0){5}}
\put(140,15){\line(1,0){45}}
\end{picture}
\]
with $m-1$ vertices.  The vertices represent the lattice basis
vectors.  The edges connect the pairs with $B(e_i,e_j) \ne 0$
and $B(e_i,e_i) = -2$ for all $i$ as above.  
In our case, the vertices represent the components $D_i$
of the exceptional divisor, and the bilinear form is the 
intersection product.  

A precise definition of a surface \emph{rational double point}  
follows.  We will say that a resolution of singularities
\[
\pi : Y \to X.
\]
is \emph{minimal} if no component of the exceptional divisor
is a curve with self-intersection number $-1$ (see Example~\ref{8.excfirst}
and the following comments).  

\begin{definition}
\label{8.ratsing}
A singular point $p$ of a surface $X$ is said to be a 
\emph{\bfseries rational double point, or Du Val singularity}
if there is a minimal resolution of singularities $\pi : Y \to X$ 
such that every irreducible component $E_i$ of the exceptional divisor $E$ over
$p$, $i = 1, \ldots, r$, satisfies
\begin{equation}
\label{8.Kinter}
(K_Y\cdot E_i) = 0,
\end{equation}
where $K_Y$ is the canonical divisor class on $Y$.
\end{definition} 

By the \emph{adjunction formula} on $Y$ (see \cite[Chapter IV]{shafarevich},
\cite[Chapter V]{hartshorne}, or \S 4 of this chapter), the
equations $(K_Y \cdot E_i) = 0$ for all $i$ imply that $E_i \simeq \P^1$ 
is a smooth curve, and $(E_i\cdot E_i) = -2$ for 
all $i$.  It follows that $E = E_1 + \cdots + E_r$ also
satisfies $(E\cdot E) = -2$.  
It also follows from this condition 
that the canonical sheaf on $Y$ is the pullback
of the canonical sheaf on $X$ under $\pi$, and this is one 
way to define \emph{rational singularities} more generally.
All toric surface singularities satisfy this last condition, but
they may not be realizable as singularities of embedded
surfaces in $\C^3$ , so we may not be able to define a multiplicity 
in the naive way we did in Example~\ref{8.rdp}.  
In Exercise~\ref{8.ratsingex},
you will verify some of these claims. 

\exercises

\small

\begin{exercise}
\label{8.smoothrefex}  
In this exercise, you will
verify several statements made in the proof of 
Lemma~\ref{8.smoothref}
\begin{thmenum}
\item Show that if the cone $\sigma$ is strictly
convex, then the integers $a_i$ in \eqref{8.lincomb}
must be strictly positive.
\item Show that if $v_{i-1} + v_{i+1} = v_i$, then 
$\Cone(v_{i-1},v_{i+1})$ must also be smooth.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{8.nega1}  In the proof of Theorem~\ref{8.smoothclass},
verify that if $v_j = -v_1$ and $v_0 = -v_2 + a_1v_1$
with $a_1 < -2$, then $\Sigma$ is a refinement of a
fan $\Sigma'$ with $X_{\Sigma'} \simeq \cH_{r}$, where
$r = |a_1|$.
\end{exercise}

\begin{exercise}
\label{8.thmex} In this exercise, you will supply 
details for several claims in the last paragraph of the proof of
Theorem~\ref{8.smoothclass}.
\begin{thmenum}
\item  Show that $v_j + v_{j+1}$ must lie in the 
third quadrant when $v_j$ is the last vector in the 
fourth quadrant, and $v_{j+1}$ lies in the second quadrant.
\item Verify the details to show
that $bc - ad > 1$ in both cases $d > 0$ and $d < 0$.
\end{thmenum} 
\end{exercise}

\begin{exercise}
\label{8.F1}
In the statement of Theorem~\ref{8.smoothclass}, you 
might have noticed the absence of the Hirzebruch surface
$\cH_1$.  Show that this surface is actually isomorphic
to the blowup of $\P^2$ at one $T_N$-fixed point.
\end{exercise}


\begin{exercise}
\label{8.nointer}
Show that if $C$, $D$ are the divisors on $X_\Sigma$
corresponding to one-dimensional cones that are not
faces of the same two-dimensional cone, then
\[
(C\cdot D) = 0.
\] 
\end{exercise}

\begin{exercise} 
\label{8.Hirzex}
This exercise studies several further
examples of the intersection product on $\cH_r$.
\begin{thmenum}
\item Check the equation $(D_3\cdot D_3) = r$ using 
Theorem~\ref{8.selfintno}.
\item Compute $(D_4\cdot D_4)$ using \eqref{8.Hirzinterprod}
and also directly from Theorem~\ref{8.selfintno}.  
\item Compute $K^2 = (K\cdot K)$ on $\cH_r$.
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{8.negdef}
Use \eqref{8.excinter} to show that the restriction of the 
intersection product to the subgroup of $\Pic(X_\Sigma)$
generated by the components of the exceptional divisor
of the resolution of a singular point $p_\sigma$ corresponding
to a cone $\sigma$ with parameters $m,k$ yields a negative-definite
associated quadratic form.  Hint: recall that if $B(x,y)$ is 
a bilinear form, the associated quadratic form is $Q(x) = B(x,x)$.
\end{exercise}

\begin{exercise}
\label{8.ratsingex}  This exercise deals with the
rational singularities from Definition~\ref{8.ratsing}.
\begin{thmenum}
\item Show that \eqref{8.Kinter} is
satisfied for the standard toric resolution of the rational double
point of type $A_{m-1}$.
\item Show that $E = E_1 + \cdots + E_{m-1}$ on the toric resolution satisfies 
$(E\cdot E) = -2$.  
\item Show that if $b_i \ge 3$ for some $i$ in the Hirzebruch-Jung
continued fraction expansion of $m/k$, then the components of the
exceptional divisor on the toric resolution do not satisfy \eqref{8.Kinter}.
\item (This part requires use of the \emph{adjunction formula} -- see \S 4 of 
this chapter.)
Show that if $C$ is an irreducible curve on a smooth surface $Y$ with $(C\cdot C) < -1$
and $(K_Y\cdot C) = 0$, then $C$ must be isomorphic to $\P^1$.  
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{8.Noether}  Another interesting numerical fact about
the integers $b_i$ from \eqref{8.lincomb} is the following.
Suppose a smooth fan $\Sigma$ has one-dimensional
cones labeled as in Lemma~\ref{8.raygens}.  Then 
\begin{equation}
\label{8.Noethereq}
b_0 + b_1 + \cdots + b_{d-1} = 3d - 12.
\end{equation}
This exercise will sketch a proof of this fact. 
\begin{thmenum}
\item Show that \eqref{8.Noethereq} holds for the standard fans 
yielding $\P^2$, $\P^1\times\P^1$, and $\cH_r$, for $r \ge 2$.
\item Show that if \eqref{8.Noethereq} holds for a smooth $\Sigma$, then it
holds for the fan obtained by performing a star subdivision
on one of the two-dimensional cones of $\Sigma$.
\item Deduce that \eqref{8.Noethereq} holds for all smooth fans using
Theorem~\ref{8.smoothclass}.
\end{thmenum}
\end{exercise}

\normalsize

\section{Riemann-Roch and Lattice Polygons}

\emph{Riemann-Roch theorems} are a class of results about the 
dimensions of sheaf cohomology groups.  
The original statement along these lines
was the theorem of Riemann and Roch concerning sections of 
line bundles on algebraic curves.  This result and its generalizations
to higher-dimensional varieties can be formulated most conveniently
in terms of the following notion of the Euler characteristic of 
a sheaf.

\begin{definition}  Let $\cF$ be a coherent sheaf on a projective
variety $X$.
The \emph{\bfseries Euler characteristic} of $\cF$, denoted
$\chi(\cF)$ is the integer
\[
\chi(\cF) = \sum_{i = 0}^{\dim(X)} (-1)^i \dim H^i(X,\cF).
\]
\end{definition}

For instance, if $X$ is a curve and $\cL$ is a line bundle on $X$,
then 
\[
\chi(\cL) = \dim H^0(X,\cL) - \dim H^1(X,\cL).
\]

One of the key facts we will need about Euler characteristics is
an additivity statement.

\begin{proposition}
\label{8.add}
Let 
\begin{equation}
\label{8.sexseq}
0 \longrightarrow \cF' \longrightarrow \cF \longrightarrow \cF'' \longrightarrow 0
\end{equation}
be a short exact sequence of coherent sheaves on a projective variety $X$.
Then 
\[
\chi(\cF) = \chi(\cF') + \chi(\cF'').
\]
\end{proposition}

\begin{proof}  This is a consequence of the long exact cohomology
sequence associated to \eqref{8.sexseq}.  You will supply the 
details in Exercise~\ref{8.sexseqex}.
\end{proof}

\subsection*{\emph{\bfseries Riemann-Roch Theorem for a Curve}}
A modern form of the original Riemann-Roch theorem for curves is 
the claim that if $\cL = \cO_C(D)$ is the line bundle corresponding
to the divisor $D$ on a smooth projective curve $C$, then 
\begin{equation}
\label{8.rr1}
\chi(\cO_C(D)) = \deg(D) + \chi(\cO_C).
\end{equation}
This equality is usually rewritten using Serre duality as in 
Chapter~\ref{coho}.  Namely, if $K$ is a canonical divisor on $C$,
then we have 
\begin{eqnarray*}
H^1(\cO_C(D)) &\simeq& H^0(\cO_C(K - D))^*\\
H^1(\cO_C) & \simeq & H^0(\cO_C(K))^*.
\end{eqnarray*}
We will call the integer $g = \dim H^0(\cO_C(K))$ the 
\emph{genus} of the curve $C$.  Then \eqref{8.rr1} can be rewritten in
the form commonly used in the theory of curves:
\begin{equation}
\label{8.rr2}
\dim H^0(\cO_C(D)) - \dim H^0(\cO_C(K - D)) = \deg(D) + 1 - g.
\end{equation}
A proof of this theorem and a number of its applications are
given in \cite[Chapter IV]{hartshorne}.  Also see Exercise~\ref{8.rrcex}
below.  As a first consequence,
note that if $D = K_C$ is a canonical divisor, then 
\begin{equation}
\label{8.candeg}
\deg(K_C) = 2g - 2.
\end{equation}

We will need to use \eqref{8.rr2} most often
in the simple case $X \simeq \P^1$.  Then $g = 0$
and the Riemann-Roch theorem for $\P^1$ is the statement
\begin{equation}
\label{8.rrP1}
\chi(\cO_{\P^1}(D)) = \deg(D) + 1
\end{equation}
for all divisors $D$ on $\P^1$.  

\subsection*{\emph{\bfseries The Adjunction Formula}}
For a curve $C$ in a smooth surface $X$, we also have a relation 
between the canonical divisors on the curve and the surface.  
From the residue exact sequence from Chapter~\ref{coho},
if $\cK_X$ is the canonical sheaf on the surface and
$\omega_C$ is the canonical sheaf on the curve (or the dualizing
sheaf if $C$ is singular), then
\[
0\longrightarrow \cK_X \longrightarrow \cK_X \otimes \cO_X(C) 
\longrightarrow \omega_C \longrightarrow 0.
\]
In other words, the sheaf $\omega_C$ on $C$ is given by
\begin{equation}
\label{8.cansheaf}
\omega_C \simeq \cK_X \otimes \cO_X(C) \otimes \cO_C.
\end{equation}
Combining this with \eqref{8.candeg} above gives the following
formula.

\begin{theorem}[Adjunction Formula] 
\label{8.adj}
Let $C$ be a smooth curve in a smooth surface $X$, then 
in the intersection product on $X$, 
\[
(K_X\cdot C) + (C\cdot C) = 2g - 2,
\]
where $g$ is the genus of the curve $C$.
\end{theorem}

When $C$ is a singular curve in a smooth surface, there is a similar 
formula, but the \emph{arithmetic genus} $p_a$ replaces the ordinary
genus $g$.  By definition,
\[
p_a = (-1)(\chi(\cO_C) - 1)
\]
for a curve $C$.
You will supply the details of the proof in Exercise~\ref{8.adjex}.

\subsection*{\emph{\bfseries Riemann-Roch Theorem for a Surface}}
The statement for surfaces corresponding to \eqref{8.rr1} is given next.

\begin{theorem}[Riemann-Roch for Surfaces]  
\label{8.rrsurf}
Let $X$ be a smooth
projective surface, let $\cO_X(D)$ be the line bundle on $X$
corresponding to a divisor $D$, and let $K$ be a canonical divisor.  
Then
\begin{equation}
\label{8.rrs}
\chi(\cO_X(D)) = \frac{(D\cdot D) - (D\cdot K)}{2} + \chi(\cO_X),
\end{equation}
where $(D\cdot D)$ and $(D\cdot K)$ are the intersection products
on $X$ as in \S 3.   
\end{theorem}

We will only prove this for $X$ a smooth complete toric surface;
there is a simple and concrete proof in this case.

\begin{proof} 
Note that \eqref{8.rrs} certainly holds if
$D \equiv 0$, so $\cO_X(D) \simeq \cO_X$.  
Our proof will use the special properties
of smooth $X$ toric surfaces in the following way.  Recall 
that if $X = X_\Sigma$, then $\Pic(X)$ is generated by the 
isomorphism classes of $\cO(D_i)$ where $D_i$, $i = 1, \ldots, d$
are the divisors corresponding to the one-dimensional cones in 
$\Sigma$.   Hence, to show that \eqref{8.rrs} holds for all $D$ 
on $X$, by a ``two-sided induction,'' 
it suffices to show that if \eqref{8.rrs} holds for a divisor 
$D$, then it also holds for $D' = D + D_i$ and $D'' = D - D_i$ 
for all $i$. 

Assume first that \eqref{8.rrs} holds for $D$,
and consider $D' = D + D_i$.  From the exact sequence
\[
0 \longrightarrow \cO_X(-D_i) \longrightarrow \cO_X \longrightarrow \cO_{D_i} \longrightarrow 0,
\]
if we take tensor product with $\cO_X(D + D_i)$ we obtain an exact sequence
of sheaves
\[
0 \longrightarrow \cO_X(D) \longrightarrow \cO_X(D+D_i) \longrightarrow \cO_{D_i}(D+D_i) \longrightarrow 0.
\]
By Proposition~\ref{8.add}, it follows that 
\[
\chi(\cO_X(D+D_i)) = \chi(\cO_X(D)) + \chi(\cO_{D_i}(D+D_i)).
\]
By the induction hypothesis, the first term on the right is
\begin{equation}
\label{8.eq1}
\chi(\cO_X(D)) = \frac{(D\cdot D) - (K\cdot D)}{2} + \chi(\cO_X).
\end{equation}
The second term can be computed using the Riemann-Roch theorem for
curves and the fact that $D_i \simeq \P^1$.  By \eqref{8.rrP1}, 
\begin{equation}
\label{8.eq2}
\chi(\cO_{D_i}(D + D_i)) = (D\cdot D_i) + (D_i\cdot D_i) + 1.
\end{equation}

Hence combining \eqref{8.eq1} and \eqref{8.eq2}, we have $\chi(\cO_X(D+D_i))$
\begin{eqnarray*}
&=& \frac{(D\cdot D) - (K\cdot D)}{2}  + (D\cdot D_i) + (D_i\cdot D_i) + 1+ \chi(\cO_X)\\
&=& \frac{((D+D_i)\cdot (D+D_i)) + (D_i\cdot D_i) - (K\cdot D) + 2}{2} + \chi(\cO_X)\\
&=& \frac{((D+D_i)\cdot (D+D_i)) - (K\cdot (D+D_i))}{2} + \chi(\cO_X),
\end{eqnarray*}
where the last equality follows since $K = -(D_1 + \cdots + D_d)$, hence
\[
(K\cdot D_i) = -(D_i\cdot D_i) - 2.
\]
Recall that $(D_i \cdot D_j)$ is $1$ when 
the corresponding one-dimensional cones in $\Sigma$ are faces of
the same two-dimensional cone and $0$ otherwise.  

The case $D'' = D - D_i$ is similar and is left to the reader as 
Exercise~\ref{8.rrex}.
\end{proof}

By the results of Chapter~\ref{coho}, for a smooth toric surface $X$ we have
\[
\chi(\cO_X) = \dim H^0(\cO_X) - \dim H^1(\cO_X) + \dim H^2(\cO_X) = 1 - 0 + 0.
\]
Moreover, in the case that $D = \sum m_i D_i$ with $m_i \ge 0$ for 
all $i$, it will be true that
\[
\dim H^1(\cO_X(D)) = \dim H^2(\cO_X(D)) = 0.
\]
Hence for toric surfaces, Theorem~\ref{8.rrsurf} gives even more direct information 
about $\dim H^0(\cO_X(D))$ than might be apparent at first glance:  For 
$D \equiv \sum m_i D_i$ with $m_i \ge 0$ for all $i$,
\begin{equation}
\label{8.rreff}
\dim H^0(\cO_X(D)) = \frac{(D\cdot D) - (K\cdot D)}{2} + 1,
\end{equation}
and the terms on the right can all be computed explicitly from the 
geometry of the fan $\Sigma$.  We will consider an example shortly.
The following statement is sometimes considered a part
of the Riemann-Roch Theorem for surfaces.

\begin{theorem}[Noether's Theorem]  
\label{8.noeth}
Let $X$ be a smooth surface, and 
let $K_X$ be a canonical divisor, then 
\[
\chi(\cO_X) = \frac{(K\cdot K) + e(X)}{12},
\]
where $e(X)$ is the topological Euler characteristic of $X$ as a
four-dimensional real manifold.
\end{theorem}

The Riemann-Roch theorems given so far have been vastly generalized
by results of of Hirzebruch and Grothendieck, and the precise relation 
of Noether's Theorem to the Riemann-Roch Theorem for surfaces
is a special case of their approach.  Hence we will postpone considering
this until Chapter~\ref{rr}.  

\subsection*{\emph{\bfseries Lattice Polygons}}
For the remainder of this section, we will indicate some of the 
ways that geometric results about toric surfaces studied so far
in this section are related to geometry and combinatorics of 
lattice polygons -- polygons which are the convex hull of a
finite number of integer lattice points in the plane.  The connection
comes from the construction of a projective toric variety from a 
polytope from Chapter~\ref{poly}.  

\begin{example}
\label{8.HirzRR}
Let $X$ be the Hirzebruch surface $\cH_2$.  We
will use the notation from Example~\ref{8.HirzPic} and Example~\ref{8.Hirzform}.
Let
\[
D = D_1 + 4D_2. 
\]
Then since $K \equiv -2D_1 - 4D_2$, it is easy to compute that
\[
(D\cdot D) = 6, \quad (D\cdot K) = -8.
\]
Hence from \eqref{8.rreff},
\[
\dim H^0(\cO_{\cH_2}(D)) = \frac{6 + 8}{2} + 1 = 8.
\]
\begin{figure}[ht]
\begin{center}
\includegraphics[height=1.6in]{Chapter8Fig4.ps}
\end{center}
\caption{The polygon corresponding to the divisor $D$ and its normal fan.}
\end{figure}
From Chapter~\ref{invertible}, \S 2, recall that divisors on $\cH_2$
can also be defined via polygons that are compatible with the standard
fan $\Sigma$ yielding the Hirzebruch surface.  Writing $v_i$ for
the ray generators of the one-dimensional cones, 
\[
v_1 = e_2,\quad v_2 = e_1, \quad v_3 = -e_2, \quad v_4 = -e_1 + 2e_2,
\]
we claim that this $D$ comes from 
the polygon $P_D$ in $M_\R$ shown in Figure 4 and
defined by the facet presentation
\begin{eqnarray*}
\langle u, v_1 \rangle & \ge & 0\\
\langle u, v_2 \rangle & \ge & 0\\
\langle u, v_3 \rangle & \ge & -1\\
\langle u, v_4 \rangle & \ge & -2.
\end{eqnarray*}
By the general results of Chapter~\ref{invertible},
\[
D \equiv 0\cdot D_1 + 0\cdot D_2 + 1\cdot D_3 + 2\cdot D_4 \equiv D_1 + 4D_2,
\]
where the last equivalence comes from the facts that
$D_4 \equiv D_2$ and $D_3 \equiv D_1 + 2D_2$.  This establishes
our claim.

This is the same as the polytope defining the rational scroll from 
Example~\ref{2.rnscroll} in Chapter~\ref{poly}.  Note that 
\[
\dim H^0(\cO_{\cH_2}(D)) = 8 = |P_D \cap M|,
\]
the number of lattice points in $P_D$.  
\eoe
\end{example}

In fact, the same toric set-up may be adapted to study lattice
points in {\it any lattice polygon}.  
Let $P \subseteq M_\R \simeq \R^2$ be any lattice polygon (two-dimensional
lattice polytope).  By the results of Chapter~\ref{poly}, \S 3, we know
that there is an associated projective toric surface $X_P$.  
As abstract varieties
\[
X_P \simeq X_{\Sigma_P},
\]
where $\Sigma_P$ is the \emph{normal fan} of $P$ in the dual space $N_\R$.  
If $\Sigma_P$ is not a smooth fan, then we can refine it as in 
Theorem~\ref{8.toricressings} to obtain a resolution of singularities 
$\pi : Y \to X_P$.  We will work on the smooth surface $Y$
if $X_P$ is not smooth in the following.

From Chapter~\ref{invertible}, the polygon
$P$ \emph{also} determines a line bundle $\cO_Y(D_P)$ and divisor (class)
$D_P$ on $X_P$ or $Y$.  Moreover, by (** ref **),
\[
H^0(\cO_Y(D_P)) = \oplus_{u \in P\cap M}\ \C \cdot \chi^u.
\]
If we combine this with the Riemann-Roch equality
\eqref{8.rreff}, then we have a formula relating the 
number of lattice points in the polygon $P$ to quantities
from the geometry of the associated toric surface $Y$:
\begin{equation}
\label{8.nolatpts}
|P \cap M| = \frac{(D_P\cdot D_P) - (K_Y\cdot D_P)}{2} + 1.
\end{equation}
In fact, both $(D_P\cdot D_P)$ and $(-K_Y\cdot D_P)$ also
have elementary ``polytopal'' interpretations in terms of the 
geometry of the polygon $P$.

We begin by noting that if $P$ is given by the facet presentation
\[
\langle u, v_i \rangle \ge -a_i,
\]
using the primitive integer inward normals $v_i$ for
each face, then the inequalities
\begin{equation}
\label{8.addK}
\langle u, v_i \rangle \ge -a_i + 1
\end{equation}
define a second polygon $Q$ which is the convex
hull of the lattice points in the \emph{interior} of $P$.
You will prove this in Exercise~\ref{8.movein}.
If $P$ comes from the divisor 
\[
D = a_1 D_1 + \cdots + a_d D_d
\]
on $Y$, then since 
\[
K_Y \equiv -(D_1 + \cdots + D_d),
\]
the inequalities \eqref{8.addK} show the first statement in the following
proposition.  We
let $\mathrm{Int}(P)$ denote the interior of $P$.

\begin{proposition}
\label{8.int}  
With notation as above, 
\[
|\mathrm{Int}(P) \cap M| = \dim H^0(\cO_Y(D + K_Y)).
\]
Moreover, when $D + K_Y$ is equivalent to an effective
divisor, 
\[
|\mathrm{Int}(P) \cap M| = \frac{(D\cdot D) + (K_Y \cdot D)}{2} + 1.
\]
\end{proposition} 

\begin{proof}  
The first equality follows from \eqref{8.addK}.  The second
is an easy consequence of the Riemann-Roch formula \eqref{8.rreff}.
You will check the details in Exercise~\ref{8.intex}.
\end{proof}

This proposition also has an interesting connection with 
the adjunction formula (Theorem~\ref{8.adj}).  If we look
at the embedded projective surface $Y \subseteq \P^m$  
corresponding to $P$,
then $D$ represents the \emph{hyperplane section} class
on $Y$, that is the divisor class containing the 
curves $Y \cap H$ for hyperplanes $H \subseteq \P^m$.  
Hence $|\mathrm{Int}(P) \cap M|$ gives what is known as the 
\emph{sectional genus} of the surface.  

\begin{example}
\label{8.secgen}
Let $P = \Delta_n = \Conv(\{ne_1,ne_2,0\})$.  Then
$Y \simeq \P^2$ in its $n$th Veronese embedding and
$D_P \equiv nL$, where $L$ is the class of a line.  The
hyperplane sections are the curves of degree $n$ in $\P^2$,
which all have (arithmetic) genus
\[
|\mathrm{Int}(\Delta_n) \cap M| = \frac{(n-1)(n-2)}{2}.
\]
You will check this assertion and another example in
Exercise~\ref{8.secgenex}.
\eoe
\end{example}

We now return to the promised polytopal interpretations
of the terms on the right in \eqref{8.nolatpts}.
Our next result gives the polytopal interpretation
of the intersection product $(-K_Y\cdot D_P)$.  Let
$\partial P$ denote the boundary of $P$ in $M_\R \simeq \R^2$.

\begin{proposition}
\label{8.bdry}
Let $D_P$ be the effective divisor corresponding to $P$.  Then
\[
(-K_Y\cdot D_P) = |\partial P\cap M|.
\]
\end{proposition}

\begin{proof}
The elementary formula
\[
|P \cap M| = |\mathrm{Int}(P) \cap M| + |\partial P \cap M|
\]
is clear.  Substituting from \eqref{8.rreff} and 
Proposition~\ref{8.int}, this becomes
\[
\frac{(D_P\cdot D_P) - (K_Y \cdot D_P)}{2} + 1 = 
\frac{(D_P\cdot D_P) + (K_Y \cdot D_P)}{2} + 1 + 
|\partial P \cap M|.
\]
Some simple algebraic manipulation yields the equation to be proved.
\end{proof}

The other term on the right in \eqref{8.nolatpts} is $(D_P \cdot D_P)/2$.  This
can be interpreted using the machinery of BKK theory as in 
\cite[Chapter 7, Theorem (5.4)]{uag}.  Recall that if $f,g$ are two 
Laurent polynomials in $\C[x,x^{-1},y,y^{-1}]$, provided
that $|\V(f) \cap \V(g) \cap (\Cstar)^2|$ is finite, 
there is an upper bound on this number which is determined by
the geometry of the \emph{Newton polygons} $P = \mathrm{NP}(f)$
and $Q = \mathrm{NP}(g)$.  Namely 
\[
|\V(f) \cap \V(g) \cap (\Cstar)^2| 
\le \mathrm{area}(P+Q) - \mathrm{area}(P) - \mathrm{area}(Q),
\]
where $P+Q$ is the \emph{Minkowski sum}.  Moreover, the 
inequality will be an equality for generic choices of
$f,g$ with these Newton polygons.  In our case if we take
two general elements $f,g$ of $\oplus_{u \in P\cap M} \C\cdot \chi^u$,
then $Q = P$ as well, and the two curves
will intersect only in a finite number of points contained
in $(\Cstar)^2$.   The intersection number will be 
\[
(D_P\cdot D_P) = 
\mathrm{area}(2P) - \mathrm{area}(P) - \mathrm{area}(P) = 2\ \mathrm{area}(P).
\]
This shows the following statement.

\begin{proposition} 
\label{8.area}
Let $D_P$ be the divisor on the smooth toric surface $Y$ corresponding
to the lattice polygon $P$.  Then
\[
(D_P \cdot D_P) = 2\ \mathrm{area}(P).
\]
\end{proposition}

We are now ready to reformulate the equation \eqref{8.nolatpts}.
Substituting from Proposition~\ref{8.bdry} and Proposition~\ref{8.area},
we obtain
\begin{equation}
\label{8.Pick}
|P\cap M| = \mathrm{area}(P) + \frac{|\partial P \cap M|}{2} + 1.
\end{equation}
This is actually a very well-known result in the theory of 
lattice polygons called \emph{Pick's Theorem}.  
Our argument leading to \eqref{8.Pick} via the Riemann-Roch Theorem
is certainly not the most elementary way of proving this statement.  
But it provides an interesting glimpse of the way toric surfaces 
and lattice polygons are related.  

\subsection*{\emph{\bfseries Reflexive Polygons and The Number 12}}

The final topic we will discuss gives another example
of an interaction between toric geometry and lattice polygons, 
following the article \cite{prv}.  Recall from 
Definition~\ref{2.reflexivedef} of Chapter~\ref{poly} that a lattice 
polytope in $N_\R$ is said to be \emph{reflexive} if its facet presentation is
\[
P = \{ m \in N_\R \mid \langle m,u_F\rangle \ge -1 \text{ for all
  facets } F\}.
\]
It follows that if $P$ is reflexive, the origin is the unique 
interior lattice point of $P$ (Exercise~\ref{2.3.reflexexiveexer} of Chapter~\ref{poly}).  
Since $a_F = 1$ for all $F$,
Exercise~\ref{2.2.dual} of Chapter~\ref{poly} implies that the dual polytope is
\[
P^\circ = \Conv(u_F \mid F \text{ facet of } P).
\]
Thus $P^\circ$ is a lattice polytope and is also reflexive
(Exercise~\ref{2.3.reflexexiveexer} of Chapter~\ref{poly}).

Up to the action of matrices in $\mathrm{GL}(2,\Z)$ and translations
by elements of the lattice, there are exactly 16 equivalence
classes of reflexive polygons in the plane, shown in Figure 5.

\begin{figure}[ht]
\begin{center}
\includegraphics[height=3.2in]{Chapter8Fig5.ps}
\end{center}
\caption{The 16 equivalence classes of reflexive lattice polygons (from \cite{prv})}
\end{figure}

The article \cite{prv} gives four different proofs of the following
result.

\begin{theorem}
\label{8.no12}
Let $P$ be a reflexive lattice polygon in $N_\R$ containing the origin.
Then
\[
|\partial P \cap N| + |\partial P^\circ \cap M| = 12.
\]
\end{theorem}

One of these proofs consists of a case-by-case verification of
the statement for each of 16 equivalence classes in Figure 5.  
You will prove the theorem this way in Exercise~\ref{8.no12ex}.
This approach is straightforward, but not very enlightening!
We will give two other proofs, both using geometry of toric surfaces.

\begin{proof}
For the first, we will need the following lemma, which describes
a way to construct a fan $\Sigma(P)$, different from the normal 
fan, from each of these polygons.

\begin{lemma}
\label{8.reflfan}
Let $P$ be a reflexive polygon with the origin as its unique
interior lattice point.  Let $\Sigma(P)$ be the complete
fan whose one-dimensional cones are spanned by the rays from
the origin to the points on the boundary of $P$.  Then
$\Sigma(P)$ is a smooth fan.  
\end{lemma}

This is easy to see for each of the polygons in Figure 5.  
Also see Exercise~\ref{8.smoothex}.

Now let $Y(P)$ be the smooth toric surface corresponding to $\Sigma(P)$,
and let $D_1, \ldots, D_d$ be the divisors corresponding
to the one-dimensional cones with ray generators $v_1, \ldots, v_d$.
Note that 
\[
d = |\partial P \cap N|.
\]
Consider the \emph{anticanonical} divisor 
\[
-K_{Y(P)} \equiv D_1 + \cdots + D_d
\]
on $Y$.  The polygon in $M_\R$ corresponding to this divisor
has facet representation,
\[
Q = \{m \in M_\R : \langle m, v_i \rangle \ge -1\ \mathrm{for\ all\ } i\}.
\]
However, since $P$ is the convex hull of the $v_i$, this means
that $Q = P^\circ$ is the dual reflexive polygon.  Hence $-K_{Y(P)}$ 
is ample.  Varieties with this property are called {\it Fano varieties}.
Smooth Fano surfaces are also called {\it Del Pezzo surfaces}.

Let $Y = Y(P)$ for one of the reflexive polygons in the 
plane.  By Proposition~\ref{8.bdry}, taking $D = -K_Y$, we have 
\begin{equation}
\label{8.K2firstway}
(-K_Y\cdot -K_Y) = |\partial P^\circ \cap M|.
\end{equation}
However, we can also compute $(-K_Y\cdot -K_Y)$ directly 
using the intersection products
\[
(D_i\cdot D_j) = \begin{cases} -b_i & \mathrm{if}\ i = j\\
                   1  & \mathrm{if}\ v_i\ \mathrm{and}\ v_j\ \mathrm{are\ adjacent}\\
                   0  & \mathrm{otherwise},
                  \end{cases}
\]
where the $b_i$ are the integers from Lemma~\ref{8.raygens}.  This computation 
shows
\begin{equation}
\label{8.K2secondway}
(-K_Y\cdot -K_Y) = -\sum_{i=1}^d b_i + 2d = 12 - d,
\end{equation}
where the last equation follows from Exercise~\ref{8.Noether}.
Since $d = |\partial P \cap N|$, combining \eqref{8.K2firstway}
and \eqref{8.K2secondway} yields
\[
|\partial P \cap N| + |\partial P^\circ \cap M| = 12,
\]
which is what we wanted to show.
\end{proof}

The second proof we will present is even simpler, though less
elementary.  

\begin{proof} 
We construct the smooth toric surface $Y$ exactly as in the first proof.
By Noether's Theorem quoted above (Theorem~\ref{8.noeth}),
\[
\chi(\cO_Y) = 1 = \frac{(K_Y\cdot K_Y) + \chi(Y)}{12}
\]
for a smooth toric surface, since $H^1(\cO_Y) = H^2(\cO_Y) = 0$.  
From \eqref{8.K2firstway} above, 
\[
(K_Y\cdot K_Y) = |\partial P^\circ \cap M|.  
\]
The topological Euler characteristic $\chi(Y)$ is the 
alternating sum of the Betti numbers (ranks of the singular
homology groups with integer coefficients). But $Y$ is 
connected, orientable, simply-connected, and satisfies Poincar\'e
duality.  Hence 
\[
\chi(Y) = 1 - 0 + (d-2) - 0 + 1 = d = |\partial P \cap N|.
\]  
Hence the desired result follows directly from Noether's Theorem.  
\end{proof}

Higher dimensional reflexive polytopes and their associated 
toric varieties are currently being actively studied.
See for instance \cite{nill}.  One reason for the interest
in these varieties is the relation with mirror symmetry in 
physics.  (**need more complete summary of literature, pointer
ahead to where this discussion will be continued (if it will!)**)

\exercises
\small

\begin{exercise} 
\label{8.sexseqex}
 Use the long exact cohomology sequence associated
to \eqref{8.sexseq} to prove Proposition~\ref{8.add}.
\end{exercise}

\begin{exercise}
\label{8.adjex}
Prove the Adjuction Formula, Theorem~\ref{8.adj}, using
\eqref{8.cansheaf} and \eqref{8.candeg}.
\end{exercise}

\begin{exercise}
\label{8.rrex}  
Complete the proof of Theorem~\ref{8.rrsurf} by showing that 
if \eqref{8.rrs} hold for $D$, then it also holds for $D'' = D - D_i$
where $D_i$ is any one of the divisors corresponding to the one-dimensional
cones in $\Sigma$.
\end{exercise}

\begin{exercise}
\label{8.rrcex}
The Riemann-Roch theorem for curves, in the form \eqref{8.rr1}, 
can be proved by much the same method as used in the previous
exercise.  Namely, show that if \eqref{8.rr1} holds for
a divisor $D$ then it also holds for the divisors $D + P$
and $D - P$, where $P$ is an arbitrary point on the curve.
\end{exercise}

\begin{exercise} 
\label{8.movein}
Let $P \subseteq M_\R$ be the polygon
defined a collection of inequalities 
\[
\langle u, v_i \rangle \ge -a_i, \quad 1\le i \le d,
\]
where the $v_i$ are a collection of primitive integer vectors 
in $M$.  Show that the inequalities
\[
\langle u, v_i \rangle \ge -a_i+1, \quad 1\le i \le d,
\]
define the convex hull of the interior lattice points in $P$.
\end{exercise}

\begin{exercise}
\label{8.intex}
Prove the second assertion in Proposition~\ref{8.int} using 
the Riemann-Roch theorem.
\end{exercise}

\begin{exercise}
\label{8.secgenex}  
This exercise studies the \emph{sectional genus} of toric surfaces
introduced in Example~\ref{8.secgen}.
\begin{thmenum}
\item Verify the formula given in the example for the sectional
genus of $\P^2$ in its $n$th Veronese embedding.
\item Let $P = \Conv(\{0,a e_1, b e_2, a e_1 + b e_2\})$.
What is the abstract smooth toric surface $Y$ in this case?  Show
that its sectional genus is $(a-1)(b-1)$.  
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{8.no12ex}  This exercise develops a brute-force
proof of Theorem~\ref{8.no12}.
\begin{thmenum}
\item For each of the polygons in Figure 5, determine the dual polygon $P^\circ$.
(Some are self-dual.)
\item Using your results from part a, verify the 
equality $|\partial P \cap N| + |\partial P^\circ \cap M| = 12$ directly. 
\end{thmenum}
\end{exercise}

\begin{exercise}
\label{8.smoothex}
In this exercise, you will show a criterion that implies
the fan from Lemma~\ref{8.reflfan} is smooth.
\begin{thmenum}
\item Show that a two-dimensional cone $\sigma = \Cone(v_1,v_2)$
in $N$ is smooth if and only if $\Conv(0,v_1,v_2) \cap N = \{0,v_1,v_2\}$.
Hint:  Show that if $e_1,e_2$ is a normalized basis of $N$ with 
respect to $\sigma$, then if $m > 1$, then $e_1$ is in 
$\Conv(0,v_1,v_2) \cap N$.  
\item Deduce that the fan $\Sigma$ from Lemma~\ref{8.reflfan} is 
smooth whenever $P$ is reflexive.
\end{thmenum}
\end{exercise}

\begin{exercise}
According to Theorem~\ref{8.smoothclass}, every smooth toric surface 
is a blow-up of either $\P^2$, $\P^1\times \P^1$, or $\cH_r$ for
$r \ge 2$.  
\begin{thmenum}
\item For each of the 16 reflexive polygons in Figure 5,
we have a smooth toric surface $Y(P)$ associated 
to the fan $\Sigma(P)$ as in Lemma~\ref{8.reflfan}.  Where
does $Y(P)$ fit in this classification in each case?  (This gives
a classification of toric Del Pezzo surfaces.)
\item From each of the polygons in Figure 5, we could also
construct a fan $\Sigma'(P)$ by taking the rays from the interior lattice
point at the origin through \emph{vertices} (but not the other
lattice points on the boundary).  How
many of the surfaces $X_{\Sigma'(P)}$ are smooth?  Identify
the ones that are.  
\end{thmenum}
\end{exercise}

\begin{exercise}
From Chapter~\ref{coho}, \S 3, recall that the 
\emph{Erhart polynomial} of a lattice polytope $P \subset M$ 
is given by 
\[
L_P(k) = |kP \cap M|,
\]
where $kP = P + \cdots + P$ (Minkowski sum).  Show that if 
$P$ is a lattice polygon in the plane, then 
\[
L_P(k) = c_2 k^2 + c_1 k + c_0,
\]
for certain rational numbers $c_2,c_1,c_0$.  Indentify these
coefficients using the geometry of $P$.
\end{exercise}
