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. "