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Algebraic Curves and Surfaces

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Algebraic Curves and Surfaces Notes for Curves and Surfaces Instructor: Robert Freidman Henry Liu April 25, 2017 Abstract These are my live-texed notes for the Spring 2017 offering of MATH GR8293 Algebraic Curves & Surfaces . Let me know when you find errors or typos. I'm sure there are plenty. 1 Curves on a surface 1 1.1 Topological invariants . 1 1.2 Holomorphic invariants . 2 1.3 Divisors . 3 1.4 Algebraic intersection theory . 4 1.5 Arithmetic genus . 6 1.6 Riemann{Roch formula . 7 1.7 Hodge index theorem . 7 1.8 Ample and nef divisors . 8 1.9 Ample cone and its closure . 11 1.10 Closure of the ample cone . 13 1.11 Div and Num as functors . 15 2 Birational geometry 17 2.1 Blowing up and down . 17 2.2 Numerical invariants of X~ ...................................... 18 2.3 Embedded resolutions for curves on a surface . 19 2.4 Minimal models of surfaces . 23 2.5 More general contractions . 24 2.6 Rational singularities . 26 2.7 Fundamental cycles . 28 2.8 Surface singularities . 31 2.9 Gorenstein condition for normal surface singularities . 33 3 Examples of surfaces 36 3.1 Rational ruled surfaces . 36 3.2 More general ruled surfaces . 39 3.3 Numerical invariants . 41 3.4 The invariant e(V ).......................................... 42 3.5 Ample and nef cones . 44 3.6 del Pezzo surfaces . 44 3.7 Lines on a cubic and del Pezzos . 47 3.8 Characterization of del Pezzo surfaces . 50 3.9 K3 surfaces . 51 3.10 Period map . 54 a 3.11 Elliptic surfaces . 56 b Chapter 1 Curves on a surface A surface for us will be smooth, projective, and connected (over C). Equivalently, it is a connected compact complex 2-fold. 1.1 Topological invariants Definition 1.1.1. There are two main topological invariants: 1. the Betti numbers bi(X) := rank Hi(X); ∼ 2. the intersection pairing H2(X; Z) ⊗ H2(X; Z) ! Z = H4(X; Z). ∼ 2 Remark. By Poincar´eduality, bi(X) = b4−i(X) and H2(X; Z) = H (X; Z), and the intersection pairing is just the cup product under this identification. Definition 1.1.2. Let H¯ 2(X; Z) := H2(X; Z)=tors. This is sensible because the torsion dies in the intersec- tion pairing, which induces a pairing H¯ 2 ⊗ H¯ 2 ! Z. Definition 1.1.3. A lattice is a free finite rank Z-module Λ together with a symmetric bilinear map (·; ·):Λ ⊗Z Λ ! Z. It is: 1. non-degenerate if Λ ! Λ_ is injective; 2. uni-modular if Λ ! Λ_ is an isomorphism. If we pick a Z-basis fα1; : : : ; αng of Λ to get the intersection matrix Aij := (αi; αj), then 1. Λ is non-degenerate iff det A 6= 0, and 2. Λ is uni-modular iff det A = ±1. Write α2 := (α; α). We say Λ is 1. type I or odd if there exists α 2 Λ such that α2 ≡ 1 mod 2, and 2. type II or even otherwise, i.e. α2 ≡ 0 mod 2 for all α 2 Λ. Accordingly, one speaks of the type or parity of the lattice. When Λ is non-degenerate, it has signature (r; s) if when A is diagonalized over R, there are r positive eigenvalues and s negative eigenvalues. Theorem 1.1.4. An indefinite unimodular lattice is characterized up to isometry by its type and signature. 1 Example 1.1.5. Clearly H¯ 2 is a free finite rank Z-module, and, equipped with the intersection pairing, is a lattice. By Poincar´eduality it is unimodular. To determine its type we use the Wu formula 2 α ≡ α · c1(X) mod 2 where c1(X) := −c1(KX ), the top Chern class of the canonical bundle. Hence 2 H¯ is of type II () c1(X) is divisible by 2: ± Let b2 (X) denote the number of positive and negative eigenvalues when the intersection pairing is diagonal- + − ized over R, so that (b2 (X); b2 (X)) is the signature. Then ¯ 2 − + H is indefinite () b2 (X) = 0 () b2(X) = b2 (X): This almost never happens. − Example 1.1.6. For smooth projective surfaces, b2 = 0 iff b2(X) = 1. In this case, b1(X) = 0, so X has the same Betti numbers as P2. Now, it does not have to be the case that X = P2, but there are only a finite number of such surfaces, and their universal covers are the unit ball in C2. 1.2 Holomorphic invariants 1 0;1 Definition 1.2.1. The most basic holomorphic invariant is the irregularity q(X) := h (X; OX ) = h (X). By Hodge theory, 0;1 1;0 0;1 b1(X) = h (X) + h (X) = 2h (X) = 2q: We say X is regular if q(X) = 0; equivalently, H1(X; R) = 0. Definition 1.2.2. Let H be a very ample divisor on X. Then there is the sheaf restriction exact sequence 0 !OX !OX (H) !OH (H) ! 0: Here OX (H) is the associated sheaf of sections of H, i.e. meromorphic functions with poles allowed along H, and OH (H) is its restriction OX (H) ⊗ OH of OX (H) to the hypersurface H. Remark. We can view OH (H) as the normal sheaf of H in X, because for divisors, OX (−H) is the conormal sheaf. In fact, this whole sequence arises simply by tensoring OX (H) onto the ideal sheaf sequence 0 ! IY !OX !OX =IY = OY ! 0. 1 1 Remark. If X is regular, then H (X; OX (H)) ! H (H; OH (H)) is surjective, because in the exact sequence, 1 H (X; OX ) = 0. Example 1.2.3. If X is a smooth surface in P3, then X is automatically regular. 0 2 Definition 1.2.4. Another holomorphic invariant is the geometric genus pg(X) := dimC H (X; Ω X) = 2 h (X; OX ) (where the equality is by Serre duality). By Hodge theory, 2;0 1;1 0;2 1;1 b2 = h + h + h = 2pg + h : Definition 1.2.5. The topological Euler characteristic is χ(X) := 1 − b1(X) + b2(X) − b3(X) + 1 1;1 = 2 − 2b1(X) + b2(X) = 2 − 4q(X) + 2pg(X) + h : The holomorphic Euler characteristic is 0 1 2 χ(OX ) := h (OX ) − h (OX ) + h (OX ) = 1 − q(X) + pg(X): 2 Theorem 1.2.6 (Hodge index theorem). Take the Hodge decomposition H2(X; C) = H2;0(X) ⊕ H1;1(X) ⊕ H0;2(X): 1. The space H2;0(X) ⊕ H0;2(X) is self-conjugate and therefore is the complexification of the real vector 2;0 0;2 space (H ⊕ H )R. On this space the intersection pairing is automatically positive-definite; 2. The space H1;1(X) is self-conjugate and therefore is the complexification of the real vector space H1;1. R There exists an element x 2 H1;1 such that x2 > 0, and the intersection form on spanfxg? in H1;1 is R R negative definite. + − 1;1 Corollary 1.2.7. b2 = 2pg(X) + 1, and b2 = h − 1. Remark. This element x is the divisor class of a hyperplane section H in a given projective embedding. In particular, H · H = d, the degree of X in the embedding. Theorem 1.2.8. Two identities involving invariants: 2 1. (Noether's formula) c1(X) + c2(X) = 12χ(OX ), which is a consequence is Hirzebruch{Riemann{Roch; + − 2 2. (Hirzebruch signature formula) b2 − b2 = (1=3)(c1 − 2c2). Remark. Any two of the Noether formula, Hirzebruch signature formula, and the Hodge index theorem imply the other. 0 ⊗n Definition 1.2.9. The plurigenera Pn(X) := dim H (X; KX ) are defined for n ≥ 1 and are \higher" holomorphic invariants of X. 1. They are not homotopy or homeomorphism invariants. 2. (Seiberg & Witten) They are diffeomorphism invariants. 1.3 Divisors Definition 1.3.1. Given a divisor D on X, there is an associated sheaf OX (D) given by OX (D)(U) := fg meromorphic on U s.t. (g) + DjU ≥ 0g; i.e. meromorphic functions with poles only along D. ∼ Remark. Note that OX (D1) = OX (D2) iff D1 ≡ D2. Remark. If DjU = V (f) and f is meromorphic on U, then OX (D)(U) = fh=f : h 2 OX (U)g so that OX (D) is a line bundle. Conversely, every line bundle on X is isomorphic to OX (D) for some D. Lemma 1.3.2. There is an exact sequence × × f1g ! C ! k(X) ! Div X ! Pic X ! 0: × Remark. If the line bundle is holomorphic, then D is effective, and 1 2 k(X) is a global section of OX (D) ∼ which vanishes along D. Conversely, if L is a line bundle and s a non-zero global section, then L = OX (D) where D = (s). 3 Definition 1.3.3. Given a divisor D, the complete linear system associated to D is × jDj := Γ(OX (D) − f0g)=C = fE effective : E ≡ Dg where ≡ is linear equivalence. The base locus of jDj is Bs(jDj) := fx 2 X : s(x) = 0 8s 2 Γ(OX (D))g = fx 2 X : x 2 E 8E 2 jDjg: We get a function N _ X n Bs(jDj) ! (P ) ; x 7! hyperplane fs 2 Γ(OX (D)) : s(x) = 0g: An effective divisor C 2 jDj is a fixed curve of jDj if E − C ≥ 0 for every E 2 jDj. 1 Remark. The analytic picture is Pic X = H (X; OX ). It is involved in the exponential sheaf sequence × 0 ! Z !OX !OX ! 0 which induces an exact sequence 1 1 1 × 2 c1 2 0 ! H (X; Z) ! H (X; OX ) ! H (X; OX ) ! H (X; Z) −! H (X; OX ) !··· where c1 is the (first) Chern class map.
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