UMD 808J: Algebraic Surfaces
1
Overview
▶
1.1
Main References
2
Positivity (and intersections)
▶
2.1
The goal
2.2
Ample line bundles
2.3
Algebraic cycles
2.4
Nakai-Moishezon
2.5
The homological nature of intersection degree
2.6
Numerical equivalence
2.7
Effective cycles and numerically effective divisors
2.8
Rational and real coefficients
2.9
Ampleness of \(\mathbb {Q}\)-divisors
2.10
Ampleness
2.11
Convex cones and ampleness
2.12
Ample and nef cones
3
Riemann-Roch and Hodge index
▶
3.1
Adjunction
3.2
Riemann-Roch for surfaces
3.3
The Hodge Index Theorem
3.4
A tiny bit more intersection theory
▶
3.4.1
Proper push-forward
3.4.2
Cycles associated to closed subschemes
3.4.3
Flat pull back
3.4.4
Localization
3.4.5
Pullback for smooth schemes and the projection formula
3.4.6
Ring structure for smooth \(X\)
3.5
Properties of the intersection pairing related to fibers
4
Blowing up and blowing down
▼
4.1
Blowing up in general
4.2
Monoidal transformations on surfaces
4.3
Strict transforms and multiplicities
4.4
Blowing Down
▶
4.4.1
Nagata’s Example
4.5
Rational maps, linear systems and indeterminacy
▼
4.5.1
Rational maps
4.5.2
Some jargon about linear systems
4.6
Elimination of indeterminacy
4.7
Harthsorne Universal Property of blowing up 2
4.8
Beauville Universal Property
4.9
Theorem on Formal Functions
4.10
Castelnuovo’s Contractibility Criterion
5
Bibliography
4.5 Rational maps, linear systems and indeterminacy
For this section, \(S\) is a smooth, projective surface (over an algbraically closed field \(k\)).