The Enriques classification of complex algebraic surfaces David Cox 02/11/2011 notes by Anna Kazanova To talk about classification of smooth surfaces in one hour is impossible, so we will talk about classification of curves and surfaces. Smooth projective curves We denote the smooth projective curve by C. We will use E for elliptic curves. Key Invariant is a genus g. We have 1 g = dim L(K ) = l(K ) = b (C); C C 2 r where KC is a canonical class divisor. Divisor D gives deg(D), L(D), l(D) = h0(D). Euler characteristic χ(D) = h0(D) − h1(D). Riemann-Roch: χ(D) = deg(D) + χ(O), where χ(O) = 1 − g. Key Results: deg KC = 2g − 2, l(KC ) = g, D is ample iff deg D ≥ 0, D is very ample iff deg D ≥ 2g + 1. Kodaira Dimension: κ(C) = supn dim φjnKC j(C). So if g = 0, then deg KC = −2 < 0, so L(nKC ) = f0g. Then κ(C) = dimf0g = −∞. If g = 1, then C is an elliptic curve, so KC ∼ 0, so L(nKC ) = C. Then κ(C) = 0. If g = 2, then deg KC > 0, so KC is ample. Then κ(C) = 1 = dim C. This case corresponds to general type. Geography and Geology • κ = −∞, g = 0, P1. 1 • κ = 0, g = 1, elliptic curve, classified by j(C) 2 C = M1 • κ = 1, g ≥ 2, general type, topology determined by g, moduli space ¯ ¯ Mg - irreducible algebraic variety of dimension 3g − 3, Mg ⊂ Mg. Mg has interesting "strata"at the boundary. Sometimes besides the curve of genus g we throw in some marked ¯ points. Mg;n ⊆ Mg;n is interesting even for g = 0. More on KX Focus only on curves of general type, i.e. g ≥ 2. Then KC is ample, but for which n is nKC very ample? First n = 3, deg(nKC ) ≥ 6g − 6 ≥ 2g + 1 (true if g ≥ 7=4). Second, n = 2, g ≥ 3, deg(2KC ) = 4g − 4 ≥ 2g + 1 (g ≥ 5=2). Third: Theorem 1.1. Let g ≥ 2. Then either KC is very ample or the 2:1 map 1 g−1 φKC : C ! P ⊆ P is hyperelliptic. Smooth Projective Surfaces Restrict to minimal surfaces, i.e. no −1–curves. We have some numerical invariants from RR. • Euler Characteristic: χ(D) = h0(D) − h1(D) + h2(D), where h2(D) = 0 h (KD). • RR: χ(D) = χ(0) + 1=2D · (D − KS) 2 • Noether Formula: χ(0) = 1=12(KS +e), where e is the topological Euler P4 i Charactristics, e = i=0(−1) bi(S). 2 2 1 We will use: c1 = KS, c2 = e, q = h (0) is called the irregularity; 0 2 pg = l(KS) = h (KS) = h (0) is called the geometric genus. So we can get from the RR: χ(0) = 1 − g + pg. Hodge Decomposition: H1(S; C) = H1;0(S) ⊕ H0;1(S), but H0;1 = H1(0) 1;0 ¯ 0;1 and H = H . Therefore b1 = 2q, or q = 1=2b1. 2 Enriques – Kodaira Classification Shafarevich: Algebraic Surfaces • κ = −∞. They all have pg = 0, q = g; 2 2 – P , c1 = 9; – Ruled Surfaces S ! C, all fibers are copies of P1, say C has genus 2 g then c1 = 8(1 − g), ∗ g = 0, Hirzebruch Surface Fn, n ≥ 2; ∗ g ≥ 1, S = P(E), where E is a rank 2 vector bundle on C. 2 • κ = 0, they all have c1 = 0 – Abelian Surfaces, (Ex. E1 × E2), c2 = 0, pg = 1, q = 2, KS ∼ 0. 3 – K3 Surface (Ex: smooth quartic in P ) c2 = 24, pg = 1, q = 0, KS ∼ 0. – Enriques Surfaces, c2 = 12, pg = 0, q = 0, KS 6∼ 0, 2KS ∼ 0. (Ex: K3/(fixed point free convolution), so we have a 2:1 covering map K3 ! Enriques, π1 = Z=2.) – Bielliptic Surface c2 = 0, pg = 0, q = 1, nKS ∼ 0, n 2 f2; 3; 4; 6g. (Ex: G acts on E1 by translations, E1=G is still elliptic, G acts on 1 E2 by automorphisms E2=G ' P . Then the surface (E1 × E2)=G is bielliptic.) Complication: Abelian and K3 can have complex versions, need not be algebraic. So complex moduli 6= algebraic moduli. 2 • κ = 1, then c1 = 0 and we have a fibration f : S ! C to a smooth projective curves C of genus g, most fibers (with finitely many exceptions) are elliptic curves, this is called an elliptic surface. (Warning: not al elliptic surfaces have κ = 1.) Furthermore, for a suitable n a map n n φjnKS jS ! P factors through C like S ! C ! P and the diagram commutes. 3 ¯ Example 1.1. Take pg ≥ 2, let n = 1 + pg, work in S ⊆ P(1; 1; 2n; 3n) with variables x; y; z; w. The equation is given by w2 = z3 + P (x; y)z + Q(x; y), where P (X; Y ) is homogeneous of degree 4n and Q(x; y) is homogeneous of degree 6n. 2 • κ = 2 then c1 ≥ 0, this is the rest! φjnKS j =? Theorem 1.2. S is a surface of general type. Then φjnKS j : S ! S¯ ⊆ PN is defined everywhere and birational if n ≥ 5 or n = 4 and 2 2 ¯ c1 ≥ 2 or n = 3, c1 ≥ 6. Furthemore, S is normal, its singularities (if any) are rational double points, and S ! S¯ is a minimal resolution of singularities, and S¯ has a singularity for every −2 curve in S. 4.