\documentclass{amsart}
\usepackage[margin=1.5cm]{geometry}
\newcommand{\Oh}{{\mathcal O}}
\newcommand{\bbA}{{\mathbb A}}
\newcommand{\bbP}{{\mathbb P}}
\newcommand{\bbQ}{{\mathbb Q}}
\DeclareMathOperator{\Coh}{{\mathrm Coh}}
\DeclareMathOperator{\rank}{{\mathrm rank}}
\DeclareMathOperator{\Hom}{{\mathrm Hom}}
\DeclareMathOperator{\Pic}{{\mathrm Pic}}
\DeclareMathOperator{\Div}{{\mathrm Div}}
\DeclareMathOperator{\NS}{{\mathrm NS}}
\newcommand{\tensor}{\otimes}
\newcommand{\into}{\hookrightarrow}
\title{Intersection Pairing}
\author{Kapil Hari Paranjape}
\begin{document}
\maketitle
We will study curves and line bundles on a smooth projective surface $S$
over a perfect field $k$.

\section{Divisors, Line Bundles and Intersection Pairing}
\begin{enumerate}
    \item The collection of line bundles $L$ (upto isomorphism) on $S$
        forms a group, with $\Oh_S$ serving as identity, tensor product
        serving as multiplication, and $L^{-1}=\Hom_S(L,\Oh_S)$ serving
        as inverse. This group is denoted as $\Pic(S)$ and called the
        Picard group of $S$.
    \item If $C$ is any reduced irreducible curve on $S$, then the ideal
        sheaf $I_{C/S}$ is a locally free sheaf of rank 1; the
        associated line bundle is denoted as $\Oh_S(-C)$.
        (This is a consequence of the fact that local rings of $S$ are
        unique factorisation domains.)
    \item Given a finite collection of curves $\{C_i\}$ and integers
        $n_i$, we call the \emph{formal} sum $D=\sum_i n_i C_i$ a
        divisor in $S$. We define a line bundle $\Oh_S(D)=\tensor_i
        \Oh_S(-C_i)^{\tensor (-n_i)}$.
    \item The collection of divisors forms an additive group denoted by
        $\Div(S)$. The above association gives a homomorphism
        $\Div(S)\to\Pic(S)$.
    \item More generally, given any closed sub-scheme $R$ of $S$ which is
        locally defined by a single equation, let $C_i$ be the reduced
        irreducible components of $R$ and net $n_i$ be the value of the
        equation defining $R$ at the generic point of $C_i$. Then
        $I_{R/S}=\tensor_i \Oh_S(-C_i)^{\tensor n_i}$. By abuse of
        notation, we can use $\Oh_S(-R)$ to denote this line bundle and
        $\Oh_S(R)$ to denote its dual. We call the associated line
        bundles and divisors \emph{effective}.
    \item Given a non-zero section $s$ of a line bundle $L$, we can
        think of $s$ as defining a homomorphism $L^{-1}\to\Oh_S$ which
        identifies $L^{-1}$ with $I_{R/S}$ where $R=Z(s)$ is the locus
        of zeroes of $s$. Thus, $L$ is isomorphic to $\Oh_S(R)$; hence,
        it is effective.
    \item A line bundle $L$ on $S$ is said to be \emph{very ample} if
        there is a closed immersion $f:S\into \bbP^n$ so that $L$ is the
        pull-back $f^{*}(\Oh_{\bbP^n}(1)$.
    \item A line bundle $L$ is said to be \emph{ample} if $L^{\tensor
        d}$ is very ample for large enough $d$.
    \item If $L$ is ample and $M$ is any line bundle (or even coherent
        sheaf), then $M\tensor L^{\tensor d}$ has a non-zero section
        for some large enough $d$. It follows that $M\tensor L^{\tensor
        d}$ is effective.
    \item Given any surjective homomorphism $M\to N$ of coherent sheaves
        in $S$ and an ample divisor $L$ on $S$, one shows that sections
        of $M\tensor L^d$ surject onto sections of $N\tensor L^d$ for
        some large enough $d$.
    \item Combining the above two results one can show that for any line
        bundle $M$ and an ample line bundle $L$, the line bundle
        $M\tensor L^d$ is very ample for all sufficently large $d$.
    \item If $C$ and $D$ are curves in $S$ meeting at a point $P$ of
        $S$, we say that the intersection is \emph{transversal} at $P$
        if the local equations $f$ and $g$ respectively of $C$ and $D$
        at $P$ generate the maximal ideal of $P$ in the local ring of
        $S$ at $P$. Note that this means that $f$ and $g$ are a regular
        system of parameters for the local ring of $S$ at $p$ and this
        $C$ and $D$ are also smooth at $P$.
    \item (Bertini's Theorem) Given any finite set $F$ of (closed)
        points of $S$ and a finite collection $\{C_i\}$ of (reduced irreducible)
        curves on $S$, and a very ample line bundle $L$ on $S$, there is
        an affine open set $U$ in the vector space $\Gamma(S,L)$ so that the
        zero locus $A=Z(s)$ in $S$ of any $s$ in $U$, satisfies the
        following conditions:
        \begin{enumerate}
            \item $A$ is a smooth irreducible curve in $S$.
            \item $A$ does not intersect (i.~e.\ contain any point of) $F$.
            \item $A$ is distinct from every curve $C_i$.
            \item $A$ meets every curve in $C_i$ transversally at every
                point lying in $A\cap C_i$.
        \end{enumerate}
    \item From what has been said above it follows that for any line
        bundle $M$ on $S$ we can find smooth curves $A$ and $B$
        satisfying the above conditions so that $M=\Oh_S(A-B)$ and the
        line bundles $\Oh_S(A)$ and $\Oh_S(B)$ are very ample. Moreover,
        we can also assume that $A$ and $B$ meet transversally (by first
        choosing one and then the other.
        In particular, we conclude that $\Div(S)\to\Pic(S)$ is a surjective
        homomorphism. 
    \item For a line bundle (or more generally a coherent sheaf) we
        define $\chi(L)=h^0(S,L)-h^1(S,L)+h^2(S,L)$ where $h^i(S,L)$ is
        the rank of the sheaf cohomology $H^i(S,L)$ as a vector space
        over $k$.
    \item If $C$ and $D$ are distinct reduced irreducible curves on $S$,
        we take the total complex of the tensor product of the complexes
        $\Oh_S(-C)\to\Oh_S$ and $\Oh_S(-D)\to\Oh_S$ To obtain a complex
        \[ 0 \to \Oh_S(-C-D) \to \Oh_S(-C)\oplus\Oh_S(-D) \to \Oh_S \]
        Since the local rings are unique factorisation domains where the
        functions defining $C$ and $D$ and distinct prime elements, we
        see that this is an exact sequence. Moreover, the cokernel at
        the last stage is the skyscraper sheaf $\Oh_{C\cap D}$. By the
        additivity of $\chi$ for terms in an exact sequece it follows
        that
        \[ \rank \Oh_{C\cap D} = \chi(\Oh_S) - \chi(\Oh_S(-C)
                    \chi(\Oh_S(-D) + \chi(\Oh_S(-C-D) \]
        Here, by abuse of notation, we are writing $\rank \Oh_{C\cap D}$
        for the rank of the global sections of this skyscraper sheaf.
    \item This leads us to define the intersection pairing of two line
        bundles as follows
        \[ (L\cdot M) =
            \chi(\Oh_S)-\chi(L^{-1})-\chi(M^{-1})
                +\chi(L^{-1}\tensor M^{-1} \]
        We also use the notation $(D_1\cdot D_2)=(\Oh_S(D_1),\Oh_S(D_2))$
        for divisors $D_1$ and $D_2$ where there is not possibility of confusion.
        We note that the formula is clearly symmetric in $L$ and $M$.

    \item The sheaf $\omega_S=\wedge^2\Omega^1_{S/k}$ is (called) the
        \emph{canonical} line bundle on $S$. A divisor $K_S$ such that
        $\Oh_S(K_S)=\omega_S$  (in other words, the divisor of zeroes and
        poles of a meromorphic 2-form on $S$) is called a \emph{canonical
        divisor}. Serre duality implies that
        \[ h^i(S,L)=h^{2-i}(S,\omega_S\tensor L^{-1}) \]
        for any line bundle $L$ on $S$. It follows that
        $\chi(L)=\chi(\omega_S\tensor L^{-1})$.
    \item A simple calculation shows that
        \[ (L^{-1}\cdot \omega_S^{-1}\tensor L) =
            2\chi(\Oh_S) - 2 \chi(L)
        \]
        We deduce the formula (sometimes called the Riemann-Roch
        formula for surfaces)
        \[ \chi(L) = \chi(\Oh_S)
                - \frac{1}{2} L\cdot (\omega_S\tensor L^{-1}) \]
        For a divisor $D$ we write this as
        \[ \chi(D) = \chi(\Oh_S(D)) = \chi(\Oh_S)
                  - \frac{1}{2} D\cdot (K_S - D) \]
    \item We can write the Riemann-Roch theorem for curves in the form
        $\chi(M)-\chi(\Oh_C)=\deg(M)$ for a line bundle $M$ on a smooth
        projective curve $C$. Suppose that $C$ lies on $S$ and that
        If $M=L_{|C}$ is the restriction to $C$ of a line bundle $L$ om
        $S$. We then have the short exact sequence,
        \[ 0 \to \Oh_S(-C) \to \Oh_S \to \Oh_C \to 0 \]
        and its tensor with $L$,
        \[ 0 \to L\tensor\Oh_S(-D) \to L \to L_{|C} \to 0 \]
        By the additivity of $\chi$ in exact sequences, we have
        \[ \deg(L_{|C}) = \chi(L) - \chi(L\tensor\Oh_S(-D)) -
        \chi(\Oh_S) + \chi(\Oh_S(-C)) \]
        In other words, $\deg(L_{|C})=-(L^{-1}\cdot\Oh_S(C))$.
        Replacing $L$ by $L^{-1}$, we obtain
        $(L\cdot\Oh_S(C))=\deg(L_{|C})$.
    \item We now define the symbol $(L_1,L_2,L_3)$ as
        \[    (L_1,L_2,L_3) = 
        L_1\cdot L_3 + L_2\cdot L_3 - (L_1\tensor L_2)\cdot L_3 \]
        If $L_3$ is $\Oh_S(C)$ for a smooth curve, then by the additivity of the
        degree of line bundles on a curve, we see that $(L_1,L_2,\Oh_S(C))=0$.
        We obtain the following symmetric expression for $(L_1,L_2,L_3)$
        by expanding all the terms
        \[ \chi(\Oh_S) - \chi(L_1^{-1}) - \chi(L_2^{-1} -
                 \chi(L_3^{-1}) 
                 + \chi(L_1^{-1}\tensor L_3^{-1})
                 + \chi(L_2^{-1}\tensor L_3^{-1}) 
                 + \chi(L_1^{-1}\tensor L_2^{-1}) 
                 - \chi(L_1^{-1}\tensor L_2^{-1} \tensor L_3^{-1})
        \]
        It follows that for any two line bundles $L$ and $M$ and any
        smooth projective curve $C$ on $S$,
        \[ (L\cdot M) = (L\tensor\Oh_S(C)\cdot M) - \deg(M_{|C}) \]
        Now, we write $L=\Oh_S(A-B)$ for suitable smooth projective
        curves $A$ and $B$ on $S$, and take $C=B$ o obtain
        \[ (\Oh_S(A-B)\cdot M) = \deg(M_{|A}) - \deg(M_{|B}) \]
        The right hand side is additive in $M$ and the line bundle $L$
        is arbitrary. It follows that $(L_1,L_2,L_3)=0$ for all line
        bundles $L_i$; equivalently, the intersection pairing is
        bi-additive.
\end{enumerate}
In summary, we have shown that intersection pairing of divisors is a
symmetric bi-additive pairing.

\section{$\bbQ$-divisors, nef, ample, big, etc.}
\begin{enumerate}
    \item We will now tensor the above groups with the rational numbers
        $\bbQ$ making them all vector spaces over $\bbQ$. We thus have a
        surjective homomorphism $\Div(S)_{\bbQ}\to\Pic(S)_{\bbQ}$ and a pairing
        $(L\cdot M)$ giving a rational number for every pair $L$ and $M$
        of elements of $\Pic(S)_{\bbQ}$.
    \item We say that a line bundle $L$  on $S$
        \emph{numerically equivalent} to 0 if $L\cdot M=0$ for every
        line bundle $M$ on $S$. Note that this is equivalent to the
        assertion that $\deg(L_{|C}=0$ for every smooth irreducible
        curve $C$ on $S$. Such line bundles form a subgroup of
        $\Pic(S)$.
    \item We now define the N\'eron-Severi group (vector space!)
        $\NS(S)_{\bbQ}$ to be the quotient of $\Pic(S)_{\bbQ}$ by the
        subspace generated by line bundles that are numerically
        equivalent to 0. We use $[L]$ to denote the class in
        $\NS(S)_{\bbQ}$ of a line bundle $L$ on $S$. By abuse of
        notation, we use $[C]$ to denote the class $[\Oh_S(C)]$. We note
        that there is a non-degenerate pairing on $\NS(S)_{\bbQ}$
        induced by the pairing introduced earlier.
    \item It is known that $\NS(S)_{\bbQ}$ is a finite dimensional
        vector space.
    \item Let $\Div(S)^{\geq 0}$ denote that subset consisting of
        non-negative integer combinations of reduced irreducible curves
        on $S$. Given $A$ and $B$ in it, and non-negative integers $a$
        and $b$, we see that $aA+bB$ also lies in it. 
    \item The \emph{cone of effective divisors} in $\NS(S)_{\bbQ}$ is the
        cone of non-negative rational linear combinations of classes
        $[C]$ of reduced irreducible curves.
    \item The \emph{cone of nef divisors} is the \emph{dual cone} of the
        cone of effective divisors. In other words, we say that $D$ is
        \emph{nef} if $D\cdot C\geq 0$ for every reduced irreducible
        curve.
    \item If $C$ and $D$ are distinct reduced irreducible curves on $S$,
        we have seen above that $C\dot D$ is the length of $\Oh_{C\cap
        D}$; in particular, it is non-negative. It follows that $C$ is
        nef \emph{unless} $C\dot C <0$. More generally, if
        $F=\sum_i a_i [C_i]$ is a non-negative linear combination
        in the cone of effective curves, then $F$ is nef \emph{unless}
        $F\cdot C_i <0 $ for some $i$.
    \item Suppose that $C$ and $C'$ are distinct curves so that
        $[C]=[C']$. We see that $[C]\cdot [C] = [C]\cdot [C']\geq 0$. It
        follows that $[C]$ is nef. In other words, a curve that
        ``moves'' is automatically nef.
    \item Suppose that $A$ is a very ample line bundle associated with
        a closed embedding $S\into\bbP^n$. We have $A\cdot
        C=\deg(A_{|C})>0$ for all curves $C$. It follows that any ample
        divisor is nef.
    \item If $L$ is any line bundle and $E$ an effective divisor, the
        $A\cdot L(-nE)\to -\infty$ as $n\to\infty$. It follows that
        $L(-nE)$ cannot be effective for large $n$ for any effective
        divisor $E$. In particular, we note that $D$ and $-D$ cannot
        both be (sub)-multiples of effective divisors.
%    \item If $L$ and $M$ are very ample line bundles then $L\tensor M$
%        is very ample (by using the Segre embedding). It follows that
%        the classes of ample line bundles form a cone in $\NS(S)_{\bbQ}$
%        which is called the \emph{cone of ample divisors}.
    \item If $D\cdot D>0$ then, by the Riemann-Roch for surfaces given
        above,
        \[ h^0(nD) +h^0(K_S-nD) \geq \chi(nD) = \chi(\Oh_S) -
            (D\cdot K_S) + n^2 (D\cdot D) \to \infty \text{~as~} n\to \infty
        \]
        It follows, for large $n$, that either $nD$ or $K_S-nD$ is effective.
        In fact, this leads to a \emph{dichotomy}:
        \begin{enumerate}
            \item Suppose that $h^0(nD)=0$ for all
                large $n$. Let $m$ be such that $K_S-mD$ is an effective
                divisor $E$. Then $h^0(E-nD)$ grows like $cn^2$ for a
                positive constant $c$. However,
                $h^0(\Oh_S(E-nD)\tensor \Oh_E)$ grows at most
                like $dn$ (since $E$ is a curve). It follows that
                $h^0(-nD)$ grows like $en^2$ for some positive constant
                $e$.
            \item On the other hand, if $mD$ is effective for some $m$,
                then as seen above $K_S-nmD$ cannot be effective for
                larger) for all large $n$. Thus, $h^0(nmD)$ grows like
                $c n^2$ for some positive constant $c$.
        \end{enumerate}
        In other words, ether there is a multiple $D_1$ of $D$ so that
        $h^0(nD_1)$ grows like a positive multiple of $n^2$ or
        $h^0(-nD)$ grows like a positive multiple of $n^2$.
    \item A divisor $D$ is called \emph{big} if $h^0(nD)$ grows like a
        positive multiple of $n^2$. What we have proved above is that if
        $D\cdot D>0$, then there is a multiple $D_1$ of $D$ (positive or
        negative) which is big. Note that some multiple of a big divisor
        is effective and so, for any ample line bundle $A$ we have
        $A\cdot D\neq 0$.
    \item We deduce the Hodge Index theorem: If $D\cdot A=0$ for some
        ample divisor $A$, then $D\cdot D < 0$.
\end{enumerate}
\end{document}
