2.7 Effective cycles and numerically effective divisors
We say that a \(k\)-cycle \(Z\in Z_k X\) is effective if \(Z = \sum a_i [V_i]\) with \(a_i\geq 0\) and \(V_i\in X_{(k)}\).
A divisor \(D\in \operatorname{\mathrm{Div}}X\) is numerically effective or nef if \(\deg D\cdot C {\gt} 0\) for all closed curves \(C\subseteq X\).
Equivalently, \(D\) is nef if \(D\cdot Z \geq 0\) for all effective \(1\)-cycles \(Z\in Z_1 X\). Here’s an obvious observation.
Suppose \(D_1\) and \(D_2\) are nef Cartier divisors and \(a_1\) and \(a_2\) are nonnegative integers. Then \(a_1 D_1 + a_2 D_2\) is also nef.
This is basically saying that the nef divisors form a convex convex cone. The only thing is that people usually talk about cones in terms of the real numbers. So let’s hold off on actually defining them until we start talking about real coefficients. It will wind up being useful to think of Propostion 2.9 in relationship to the following slightly less obvious proposition.
Suppose \(D_1\) and \(D_2\) are ample (resp. very ample) Cartier divisors and \(a_1\) and \(a_2\) are nonnegative integers, which are not both \(0\). Then \(a_1 D_1 + a_2 D_2\) is also ample (very ample).
It suffices to consider the case when \(a_1\) and \(a_2\) are both positive, as the proposition is obvious otherwise.
For the case of very ample divisors, suppose \(D_1\) and \(D_2\) are very ample. Then we have embeddings \(f_i:X\to \mathbb {P}^{N_i}\) corresponding to \(D_i\). And we have the Segre embedding \(s:\mathbb {P}^{N_1}\times \mathbb {P}^{N_2}\to \mathbb {P}^{M}\) with \(M = N_1N_2 + N_1 + N_2\) and \(s^*\mathcal{O}_M(1) = \mathcal{O}_{\mathbb {P}^{N_1}\times \mathbb {P}^{N_2}} (1,1)\). But then \([s\circ (f_1\times f_2)]^*\mathcal{O}_{\mathbb {P}^M}(1) = \mathcal{O}_X(D_1+D_2)\).
For the case of ample divisors, we can either use the very ample case or prove it directly using the global generation criterion.
A class in the Chow group \(A_k X\) is effective if it can be represented by an effective cycle in \(Z_k X\).
Let’s write \(Z_k^{\operatorname{\mathit{eff}}} X\) for the group of effective \(k\)-cycles and \(A_k^{\operatorname{\mathit{eff}}}\) for the group of effective Chow classes. So \(A_k^{\operatorname{\mathit{eff}}} X\) is the image of \(Z_k^{\operatorname{\mathit{eff}}} X\) under the canonical surjection \(Z_k X\to A_k X\).
Recall that, if \(L\) is a line bundle on \(X\), then we get map \(c_1(L):A_k X\to A_{k-1} X\). Here’s a fact that follows from the way Fulton sets up intersection theory with line bundles [ 3 , Chapter 2 ] .
Suppose \(D\) is an effective Cartier divisor on \(X\) with associated line bundle \(L\) and \(V\) is a \(k\)-dimensional subvariety of \(X\) not contained in the support of \(D\). Then \(c_1(L)[V]\) is effective.
Since \(D\) is effective, there is a section \(s\in \operatorname{\mathrm{H}}^0(X,L)\) with zero locus \(D\). Then, restriction to \(V\), we get a line bundle \(L_{|V}\) and a section \(s_{|V}\in \operatorname{\mathrm{H}}^0(V,L_{|V})\). Since \(D\) does not contain \(V\), the section \(s_{|V}\) does not vanish identically on \(V\). So it defines an effective Cartier divisor \(D_V\) on \(E\) whose support is \(D\cap V\). Viewing \(D_V\) first as a Weil divisor on \(V\) and then as a cycle on \(X\), we an effective \(k-1\)-cycle on \(X\) which, by the construction in [ 3 ] is \(c_1(L)[V]\).
Suppose \(L\) is a very ample line bundle. Then the map \(c_1(L):A_k X\to A_{k-1} X\) sends \(A_k^{\operatorname{\mathit{eff}}} X\) to \(A_{k-1}^{\operatorname{\mathit{eff}}} X\).
Suppose \(V\) is a \(k\)-dimensional subvariety of \(X\). Since \(L\) is very ample, we can find a divisor \(H\) not containing \(V\) such that \(\mathcal{O}_X(H) = L\). Then use Proposition 2.12.