Intersection numbers and positivity

Intersection numbers of line bundles on proper schemes, defined through Snapper's polynomial of Euler characteristics, with multilinearity, reduction to components and divisors, degrees on curves, pullback and positivity for ample bundles; the Nakai-Moishezon criterion for ampleness; nef line bundles and Kleiman's theorem that they have nonnegative intersection numbers, so that nef plus ample is ample; and Bézout's inequality for the isolated zeros of n polynomials in n variables, with a weighted version.

Four lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Intersection numbers of line bundles
2. The Nakai–Moishezon criterion
3. Nef line bundles and Kleiman's theorem
4. Bézout's inequality for isolated zeros

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Completion (Commutative algebra for geometry)
• Dimension theory of Noetherian local rings (Commutative algebra for geometry)
• Regular sequences, depth and Cohen–Macaulay modules (Commutative algebra for geometry)
• Ample invertible sheaves · Morphisms of schemes
• Dimension of fibres · Morphisms of schemes
• Very ample invertible sheaves, Segre and Veronese embeddings · Morphisms of schemes
• Zariski's Main Theorem (Morphisms of schemes)
• Euler characteristics and Hilbert polynomials · Cohomology of quasi-coherent sheaves
• Coherence of higher direct images under proper morphisms · Cohomology of quasi-coherent sheaves
• Coherent sheaves on projective schemes: Serre's theorems · Cohomology of quasi-coherent sheaves
• Projective space, relative Proj and maps to projective space · Sheaves and schemes
• Coherent cohomology on projective schemes (course Cohomology of quasi-coherent sheaves), ampleness and morphisms (course Morphisms of schemes), and Cohen-Macaulay rings and completion (course Commutative algebra).