CW Complexes for Complex Algebraic Surfaces

CW Complexes for Complex Algebraic Surfaces

We describe CW complexes for complex projective algebraic surfaces in the context of practical computation of topological invariants. CW COMPLEXES FOR COMPLEX ALGEBRAIC SURFACES

Complex Algebraic Surfaces Class 14

Useful proposition. Consider the blow-up of P at n general points, giving exceptional divisors E1, : : : , En. Then the intersection ring on P is given by Z[H;E1; : : : ; En]=H 2 = 1; HEi = 0; Ei Ej = 0; E 2 i = 1: We can understand divisors and sections of divisors in terms of divisors on P with certain multiplicities at the Ej . More precisely: the vector space of sections of aH b1E1 bnEn is ...

Complex Algebraic Surfaces Class 6

I realize now that I am starting to be sloppy about notation for divisors and their corresponding invertible sheaves. For example, roman characters usually refer to divisors, e.g. D, and calligraphic characters usually refer to sheaves. The main cause of confusion is that we use additive notation for divisors, and multiplicative notation for sheaves. I’ll likely continue this confusion, so plea...

Complex Algebraic Surfaces Class 18

It makes use of the last general construction commonly used in the theory of algebraic surfaces, the Albanese variety. After seeing this, I think you’ll be fully equipped to read much of the literature on surfaces in the algebraic category. On Friday, the last day of class, I’ll then sketch the rest of the classification of algebraic surfaces. In particular, you’re familiar with all the ingredi...

Complex Algebraic Surfaces Class 16