Intersection numbers, ampleness criteria and Bézout's inequality, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Snapper's polynomial — Theorem 1.2. For coherent F on a proper k-scheme and invertible L_1, ..., L_r, chi(F tensor L_1^n_1 ... L_r^n_r) is a polynomial of total degree at most dim Supp F.
• Rules for intersection numbers — Proposition 2.3. Intersection numbers are integers, symmetric and multilinear, vanish on sheaves of smaller dimension, are additive, and equal the sum over top-dimensional components weighted by generic lengths.
• Restriction to a divisor — Proposition 2.4. For a section s of L_1 on V that is injective on O_V, (L_1 L_2 ... L_d . V) = (L_2 ... L_d . D), D the zero scheme of s.
• Degrees on curves — Corollary 2.5. On an integral curve, (L . C) = chi(L|C) - chi(O_C), and it equals the length of the zero scheme of any nonzero section; in particular it is nonnegative if L|C has a nonzero section.
• Pullback of intersection numbers — Proposition 3.1. (f^*L_1 ... f^*L_d . W) = [k(W):k(f(W))] (L_1 ... L_d . f(W)) if dim f(W) = dim W, and 0 otherwise.
• Zeros and poles on a curve — Corollary 3.3. A nonzero rational function on a normal integral proper curve has as many zeros as poles, with multiplicity; a nonconstant one has at least one zero and one pole.
• Positivity of ample bundles — Theorem 3.4. (L^d . Z) > 0 for L ample and every closed subscheme Z of dimension d.
• Hyperplane sections — Proposition 3.5. For H very ample and V integral of dimension d, (H . L_2 ... L_d . V) is a positive integer combination of intersection numbers on integral subvarieties of dimension d - 1.
• Globally generated and positive on curves — Lemma 1.3. A globally generated invertible sheaf on a proper k-scheme with positive degree on every curve is ample.
• The Nakai-Moishezon criterion — Theorem 2.1. On a projective scheme over an algebraically closed field, L is ample if and only if (L^dim V . V) > 0 for every integral closed subscheme V of positive dimension.
• Kleiman's theorem — Theorem 2.1. If L is nef and A ample on a projective scheme, then (L^i . A^(d-i) . V) >= 0 for every integral V of dimension d; in particular (L^d . V) >= 0.
• Nef plus ample is ample — Corollary 3.1. If L is nef and A ample, then aL + bA is ample for all integers a >= 0, b >= 1.
• Rational combinations — Corollary 3.3. If cL = aN + bM with N nef, M ample and positive integers a, b, c, then L is ample.
• Macaulay's theorem — Theorem 1.2. If the leading forms of f_1, ..., f_n in F[x_1, ..., x_n] have no common nontrivial zero, then dim F[x]/(f) = deg f_1 ... deg f_n.
• Isolated zeros in a deformation — Proposition 2.1. At an isolated zero P of the special fibre of a one-parameter deformation of n equations, the completed local algebra is free of rank mu_P over k[[t]].
• Bézout's inequality — Theorem 3.1. For g_1, ..., g_n in k[x_1, ..., x_n] with deg g_i <= d_i (k algebraically closed), the multiplicities of the isolated zeros add up to at most d_1 ... d_n.
• Weighted Bézout inequality — Corollary 3.2. In characteristic zero, for polynomials of weighted degree at most N with weights rho_1, ..., rho_n, the multiplicities of the isolated zeros add up to at most N^n/(rho_1 ... rho_n).

Reading order

Read the first lesson for intersection numbers and their rules. The second proves the Nakai-Moishezon criterion and the third Kleiman's theorem on nef bundles; together they show that nef plus ample is ample. The fourth lesson is independent of the others and proves Bézout's inequality for isolated zeros, including the weighted form used in zero estimates.