Week 3 Cartier divisors, Weil divisors and intersection numbers on surfaces
Main reference: Mumford, Lectures on curves on a surface, Chapter 9. My
Chapter A, Shafarevich
Summary: Cartier divisors on Noetherian schemes lead to treatment of
Pic as divisor up to linear equivalence. For a normal integral
scheme there is a rival notion of Weil divisor. However, Cartier
divisors have many virtues that Weil divisors lack: (1) intersection
numbers, (2) contravariant functoriality, (3) compatibility of
divisorial sheaves with tensor product.
Blow up one point
"abstract local" versus "global projective" construction
Single blowup of PP^2 = projection of Veronese surface
= del Pezzo -- Segre cubic scroll FF1
Blow up at 3 general points and dP6. Famous because of the standard
quadratic Cremona transformations.
Characterisation of blow-up and -1-curve in a general setting.
Traditional birational statement of the minimal model of surfaces.
Statement of Main Proposition on C.D
discussion of geometric case and l(OX/(f,g))
discussion of D1.D2 < 0.
Theoretical advantage of using Cartier divisor and thinking of the
intersection pairing as bilinear map N^1 x N_1 -> ZZ between two
different spaces.
Relation between D1.D2 and cup product in cohomology: the inclusion
of N^1/numeq as c_1 (of line bundles) in H^2.
N_1 in H_2(X, ZZ) and H^2 ->> N^1
Néron-Severi group in char 0, and what you have to do to prove the
same over a general field. See the Wikipedia page:
"In algebraic geometry, the Néron–Severi group of a variety is the group
of divisors modulo algebraic equivalence; in other words it is the group
of components of the Picard scheme of a variety. Its rank is called the
Picard number. It is named after Francesco Severi and André Néron. "