Intersection Numbers of Varieties

We adopt an alternative definition of Cartier divisors than seen for example in Vakil's FOAG. Let $X$ be a scheme, and define the sheaf of total quotient rings $\fank_X$ as the sheaf whose $U$-points $\fank_X(U)$ is the localization of $\ox(U)$ by the multiplicative set of non zero divisors. We can define a Cartier divisor as a global section of the sheaf $\fank_X^*/\ox^*$, where $\fank_X^*(U)$ is the group of invertible elements in $\fank_X(U)$.

Perhaps more concretely, a Cartier divisor is a collection of tuples $(U_i, f_i)$ where $U_i$ is an open cover of $X$ and $f_i$ is a unit in $\fank_X(U_i)$, such that for any other tuple $(U_j, f_j)$ we have $f_i/f_j \in \calo_X^*(U_i \cap U_j)$.

Note that if $X$ is reduced then we may take a cover by integral (=reduced + irreducible) $U_i$ such that $\fank_X(U_i) = K(U_i)$ is just the field of fractions of $\ox(U_i)$. Then $f_i$ is merely a nonzero rational function on $U_i$ such that $f_i/f_j$ is a nonvanishing regular function on $U_i \cap U_j$.

We say a Cartier divisor is principal if it is the image of the natural map

That is, it can be defined by a global nonzero rational function on $X$. Two Cartier divisors $D,D'$ are linearly equivalent if $D-D'$ is principal, in which case we write $D \equiv D'$. We say $D$ is effective if for all tuples $(U_i, f_i)$, the $f_i$ are not zero divisors in $\ox(U_i)$; the $f_i$ cut out a codimension $1$ subscheme of $X$ whose reduction is the support of $D$.

This means we have a natural bijection between effective Cartier divisors $D$ and subschemes of $X$ which are locally cut out by a single non zero divisor. For this reason, effective Cartier divisor may refer to either the algebraic or geometric definition depending on context.

Hilbert Functions

Fix a field $k$. For this section we will assume all schemes are finite type over $k.$

Let $X$ be a proper scheme and $\fanf$ a coherent sheaf on $X.$ Recall the Euler characteristic is defined as the alternating sum

Let $X$ be a subscheme of $\bbp^N_k$ of dimension $n$ (which is projective hence proper), and recall we have the Serre twisting sheaf $\ox(m)$ restricted to $X.$ For a more general coherent sheaf $\fanf$, we defined $\fanf(m) := \fanf \otimes \ox(m).$ Then the function

is a polynomial of degree $\dim \supp \fanf$, called the Hilbert function of $\fanf.$ By Serre vanishing, this implies

is a polynomial for $m \gg 0$ called the Hilbert polynomial of $\fanf.$ In particular, if $\fanf = \ox$ then we define $p_X(m) = p_{\ox}(m)$ as the Hilbert polynomial of $X$, which has degree $\dim X$.

We make a somewhat unintuitive definition of the degree of $X$, which is $n!$ times the leading coefficient of $p_X(m)$. That is, $p_X(m) = (\deg(X)/n!) m^n + \ldots$ We can give some justification as for why this is a good notion of degree.

1. The degree of a hypersurface $H$ in $\bbp^N_k$ is the degree of the single nonzero homogeneous polynomial that cuts out $H.$ Show that this definition coincides with the above.
2. Suppose a regular curve $C$ is embedded in $\bbp^N_k$ via an invertible sheaf (line bundle) of degree $d$. Show that the degree of the embedded curve with the above definition is also $d$.

Now we return to the world of Cartier divisors and attempt to define the intersection product. Let $D_1, \ldots, D_r$ be Cartier divisors on a proper scheme $X,$ and let $\fanf$ be a coherent sheaf on $X.$ Consider the generalized Hilbert function

Then this function takes the same values on $\bbz^r$ as a polynomial with rational coefficients of degree at most $\dim \supp \fanf$.

Throughout the proof we will use the fact that, for a short exact sequence

if the result holds for $\fanh, \fang$ then it holds for $\fanf$ to reduce the problem to simpler statements.

Any coherent sheaf $\fanf$ on a Noetherian scheme $X$ admits a filtration

such that $\fanf_i/\fanf_{i+1}$ is torsion free on an integral subscheme of $X.$ We'll give a slightly nonrigorous justification. Let $\xi$ be an associated point of $\fanf$ and suppose $Z = \overline{\{\xi\}}$ is an integral closed subscheme of $X$ which is the closure of $\xi.$ Since $\supp \fanf$ is the closure of the (finitely many) associated points of $\fanf$, $Z$ must be contained in the support. The embedding $Z \to X$ yields a short exact sequence

of coherent sheaves. Let $\fanf_1 := \fani_{X/Z} \otimes \fanf$ such that $\fanf/\fanf_1 = \fanf\vert_Z.$ Since $\fanf_1$ is the sheaf of sections of $\fanf$ that are annihilated by $\xi,$ this quotient is torsion free along $Z.$ This also implies $\supp \fanf_1$ is strictly contained in $\supp \fanf,$ so by the Noetherian condition this filtration must be finite.

Back to the Theorem at hand. The existence of the filtration implies it suffices to show the result for $X$ integral and $\fanf$ torsion-free. Then there exists a dense open $U$ of $X$ over which $\fanf$ is free of rank $r$. The choice of isomorphism $\fanf \otimes \calo_U \simeq \calo_U^{\oplus r}$ determines an embedding of the following form.

Let $\fang = \fanf \cap \ox^{\oplus r}$ and note that in the sequences

the sheaves $\fang_1, \fang_2$ are not supported on $U$, hence their supports are strictly contained in $X$. By induction on the dimension of $X$, we can reduce to the case $\fanf = \calo_X$ and $X$ integral.

The rest of the proof can be found in Debarre's Higher-Dimensional Algebraic Geometry, but the gist is that this is a generalization of the fact that the Hilbert function in one variable is a polynomial. That is,

is a polynomial where $P \in \bbq[T_1,\ldots,T_r]$. Let $d$ be the total degree of $P$, and let $n_1,\ldots,n_r$ be integers such that

still has degree $d$. Then since $Q(m) = \chi(X,m(n_1D_1 + \ldots + n_r d_r))$ it follows that $d \leq \dim X$.

Intersection Numbers

The intersection number

of a collection of $r \geq \dim X$ Cartier divisors $D_1,\ldots,D_r$ on a proper scheme $X$ is defined as the coefficient of $m_1 \cdots m_r$ in the polynomial

In particular, we can show that this number is an integer and that it vanishes for $r > \dim X$. If the $D_i$ are effective and meet properly in a finite number of points, it has a nice geometric interpretation as the number of points contained in the intersection $D_1 \cap \ldots \cap D_r$ counted with multiplicity.

For $Y$ a subscheme of $X$, we can define

and similarly for $\fanf$ a coherent sheaf on $X$ with $\dim \supp \fanf \leq r$, we can define

as the coefficient of $m_1 \cdots m_r$ in