\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}