# Algebraic geometry

## Extracted Annotations

Annotations(6/1/2022, 12:47:15 PM)

• we treat two special classes of surfaces, the ruled surfaces, and the nonsingular cubic surfaces in P°, (Hartshorne, 2008, p. 373)

• As applications we give the Hodge index theorem and the Nakai-Moishezon criterion for an ample divisor. (Hartshorne, 2008, p. 373)

• we prove the theorem of factorization of a birational morphism i (Hartshorne, 2008, p. 373)

• prove Castelnuovo9s criterion for contracting an exceptional curve of the first kind. (Hartshorne, 2008, p. 373)

• Here the theory of curves gives a good handle on the ruled surfaces, because many properties of the surface arc closely related to the study of certain linear systems on the base curve. (Hartshorne, 2008, p. 373)

• there is a close connection between ruled surfaces over a curve C and locally free sheaves of rank 2 on C, (Hartshorne, 2008, p. 373)

• e study the nonsingular cubic surfaces in P3, and the famous 27 1ineS which lie on those surfaces. By representing the surface as a P° with 6 points blown up, the study of linear systems on the cubic surface is reduced (Hartshorne, 2008, p. 373)

• to the study of certain linear system of plane curves with assigned base points. (Hartshorne, 2008, p. 374)

• he Riemann4 Roch theorem for surfaces gives a connection between the dimension of a complete linear system |D|, which is essentially a cohomological invariant, and certain intersection numbers on the surface. (Hartshorne, 2008, p. 374)

• surface will mean a nonsingular projective surface over an algebraically closed field k. (Hartshorne, 2008, p. 374)

• curve on a surface will mean any effective divisor on the surface. In particular, it may be singular, reducible or even have multiple components. A point will mean a closed point, unless otherwise specified. (Hartshorne, 2008, p. 374)

• If C and D are curves on X, and if Pe C n D is a point of intersection of C and D, we say that C and D meet transversally at P if the local equations f,g of C,D at P generate the maximal ideal mp of Op x. (Hartshorne, 2008, p. 374)

• PrOOF. We embed X in a projective space P" using the very ample divisor D. (Hartshorne, 2008, p. 375)

• Then we apply Bertini9s theorem (Hartshorne, 2008, p. 375)

• Here, of course, .#(D) is the invertible sheaf on X corresponding to D (I, §7), and deg. denotes the degree of the invertible sheaf ¥ (D) ® (¢ on C (IV,§1). (Hartshorne, 2008, p. 375)

• #(4 D) is the ideal sheaf of D on X. (Hartshorne, 2008, p. 375)

• f C and D are curves with no common irreducible component, and if P ¬ C n D, then we define the intersection multiplicity (C.D)p of C and D at P to be the length of (. x/( f.¢), where f,g are local equations of C.D at P (I, Ex 5.4). Here length is the same as the dimension of a k-vector space. (Hartshorne, 2008, p. 377)

• Intersection number as a sum of local intersection multiplicities

• If D is any divisor on the surface X, we can define the selfintersection number D.D, usually denoted by D*. Evenif Cisa nonsingular curve on X, the self-intersection C? cannot be calculated by the direct metho (Hartshorne, 2008, p. 377)

• How to compute self-intersection numbers as the degree of the normal sheaf $${\mathcal{N}}_{C, X}$$.

• e must use linear equivalence. (Hartshorne, 2008, p. 378)

• Intersection numbers on P^2

• Intersection numbers of quadric surfaces in P^3

• Self-intersection of the canonical