Universita` degli Studi di Pisa Dipartimento di Matematica Corso di Laurea Magistrale in Matematica Asymptotic syzygies of algebraic varieties Tesi di Laurea Magistrale Relatore: Controrelatore: Prof. Giorgio Ottaviani Prof. Andrea Maffei Candidato: Daniele Agostini Anno Accademico 2012/2013 Contents Introduction 3 Ringraziamenti 5 1 Minimal free resolutions and Koszul cohomology6 1.1 Preliminary results......................................6 1.1.1 Spectral sequences..................................6 1.1.2 Schur functors....................................7 1.2 Algebra.............................................9 1.2.1 The Koszul complex................................. 12 1.2.2 Betti tables....................................... 16 1.2.3 Koszul cohomology as a functor.......................... 21 1.3 Geometry............................................ 22 1.3.1 Minimal free resolutions of coherent sheaves................... 23 1.3.2 Koszul cohomology of projective varieties.................... 24 2 Techniques of Koszul cohomology 25 2.1 Syzygy bundles........................................ 25 2.2 Lefschetz theorem....................................... 29 2.3 Duality............................................. 30 2.4 Castelnuovo-Mumford regularity.............................. 32 2.5 Vanishing theorems...................................... 36 2.6 Property Np .......................................... 38 2.6.1 Arithmetically Cohen-Macaulay embeddings.................. 40 2.6.2 Property Np for curves of high degree...................... 40 2.6.3 Property Np for Veronese embeddings...................... 41 2.7 Examples of Betti tables................................... 42 2.7.1 Rational normal curves............................... 43 2.7.2 Elliptic normal curve................................. 45 2.7.3 Veronese surface................................... 46 2.7.4 Higher degree Veronese surfaces.......................... 48 3 Asymptotic syzygies of algebraic varieties 49 3.1 Asymptotic Betti tables.................................... 49 3.1.1 The case of Kp,1 .................................... 51 3.2 Secant constructions..................................... 52 3.3 The asymptotic non-vanishing theorem.......................... 57 3.3.1 Preliminary constructions.............................. 57 3.3.2 The proof....................................... 60 3.4 The asymptotic non-vanishing theorem for Veronese varieties............. 62 1 4 Asymptotic normality of Betti numbers 64 4.1 Asymptotic Betti numbers of rational normal curves.................. 64 4.1.1 Discrete random variables and generating functions.............. 65 4.1.2 Asymptotic normality for Betti numbers of rational normal curves...... 67 4.2 Asymptotic normality for Betti numbers of smooth curves............... 69 5 Cohomology of homogeneous vector bundles 73 5.1 Notations and preliminaries................................. 73 5.2 Vector bundles and representations............................ 78 5.2.1 A distinguished open cover of G/P ....................... 78 5.2.2 Homogeneous vector bundles and representations............... 79 5.3 Hermitian symmetric varieties and Higgs bundles.................... 81 5.3.1 The gr functor..................................... 81 5.3.2 Extending representations.............................. 82 5.3.3 Higgs bundles on Hermitian symmetric varieties................ 83 5.4 Bott-Borel-Weil Theorem................................... 84 5.4.1 Bott Theorem on projective space......................... 85 5.4.2 Bott Theorem on Hermitian symmetric varieties................. 87 5.5 Quivers and relations..................................... 89 5.5.1 Basic definitions................................... 89 5.5.2 The quiver associated to an homogeneous variety................ 90 5.6 Cohomology of homogeneous vector bundles...................... 93 5.7 Example: the projective line................................. 99 5.7.1 Plethysm of ^n(Sm(V)) ............................... 100 6 The Veronese surface 103 6.1 Working on the projective plane.............................. 103 6.1.1 The case d = 2.................................... 104 6.1.2 The general case................................... 106 6.2 Working on the flag variety................................. 111 6.2.1 Homogeneous bundles on the flag variety.................... 111 6.2.2 Cohomology and Random triangles........................ 113 References 121 2 Introduction The purpose of this thesis is the exposition of some recent results about syzygies of projective varieties. More specifically, consider an algebraically closed field k of characteristic zero, and a smooth connected projective variety X over k. Let L be a very ample line bundle on X inducing a projectively normal embedding X ,−! P(H0(X, L)) = Pr and let S = S•(H0(X, L)) be the homogeneous coordinate ring of the projective space Pr and L 0 q r SX = q2Z H (X, L ) be the homogeneous coordinate ring of the embedded variety X ⊆ P . Then the (extended) minimal free resolution of SX is the unique shortest possible exact sequence 0 −! Fs −! Fs−1 −! ... −! F1 −! F0 −! SX −! 0 where every Fp is a finitely generated free graded S-module, called the module of p-th syzygies of X in Pr. We can write M Fp = S(−p − q) ⊗k Kp,q(X, L) q2Z for certain uniquely determined finite-dimensional vector spaces Kp,q(X, L). The numbers def kp,q(X, L) = dimk Kp,q(X, L) are called the graded Betti numbers of X and they encode many of the algebraic and geometric properties of the variety. ⊗d In this work, we are interested in studying the Betti numbers kp,q(X, L ) as d grows to infinity. The first natural question to ask is about the vanishing of these numbers. From an intuition ⊗d based on results about smooth curves, one could think that the kp,q(X, L ) would become more sparse as d increases. Instead, in their recent article [EL12], L. Ein and R. Lazarsfeld proved that ⊗d the Betti numbers kp,q(X, L ) become asymptotically nonzero for almost all possible values of p, q, moreover, they give a precise range of nonvanishings for the case of Veronese embeddings n kp,q(P , OPn (d)). In the thesis, we explain these results, presenting their proof. The next natural question to ask is about the actual values of the Betti numbers. In particular, inspired by the article [EEL13] of L. Ein, D. Erman and R. Lazarsfeld, we prove that the Betti numbers of smooth curves have an asymptotically normal behavior. More precisely, if X is a smooth curve, then we define a discrete random variable Xd with distribution ⊗d kp,1(X, L ) P(X = p) = for all p ≥ 0 d +¥ ⊗d ∑h=0 kh,1(X, L ) and then we prove the following result. d d Theorem 1. As d ! +¥ it holds that E[Xd] ∼ 2 , Var[Xd] ∼ 4 and moreover Xd − E[Xd] p −! N (0, 1) Var[Xd] in distribution. 3 Following a conjecture of Ein, Erman and Lazarsfeld [EEL13], we expect that the above result extends to higher dimensional varieties as well. Then, the next simplest case to consider would be 2 quite naturally that of Betti numbers of plane Veronese embeddings kp,q(P , OP2 (d)). In this case, it is much more difficult to get an hold on the Betti numbers with the methods used for curves, since the result of Ein and Lazarsfeld [EL12] tells us that the Betti table of 2 (P , OP2 (d)) is very non-sparse as d ! +¥. Thus, we need another technique to compute these numbers: the key point is that we can look at P2 as an SL(3)-homogeneous variety, and then the Betti numbers are given as the cohomology of certain SL(3)-homogeneous bundles on P2. The cohomology of irreducible homogeneous bundles on an homogeneous projective variety X = G/P is described by the classical Bott’s Theorem and it is quite simple. In the thesis we present a result, due to G. Ottaviani and E. Rubei [OR06], that extends Bott’s Theorem to every homogeneous bundle, exploiting an equivalence between these bundles and representations of a certain quiver. It is quite difficult to implement directly this method in our situation, since the quiver maps become quickly very complicated, but, under an additional assumption on the SL(3)-morphisms that is satisfied in all known cases, the problem is reduced to a more tractable combinatorial statement about representations of SL(2). We do not get to a proof of the asymptotic normality, but through this strategy we are able to give some partial results and we can write an algorithm 2 that computes the Betti numbers kp,q(P , OP2 (d)) in further cases than existing ones, albeit under the additional hypothesis. Turning to the contents of the single chapters, in Chapter 1 we give the basic definitions and first results relative to minimal free resolutions and Betti numbers in an algebraic and geometric context, introducing the language of Koszul cohomology. In Chapter 2 we investigate further some other aspects of Betti numbers. In particular we discuss syzygy bundles, Castelnuovo-Mumford’s regularity, duality, a Lefschetz-type theorem, property Np and we conclude by giving examples of Betti numbers for rational normal curves, elliptic normal curves and Veronese surfaces of low degree. Chapter 3 is devoted to the exposition of Ein and Lazarsfeld’s results, following their article [EL12]. In Chapter 4 we present the proof of asymptotic normality for the Betti numbers of smooth curves. To this end, we generalize the computations done for elliptic normal curves in order to get control on almost every Betti number and then we use some combinatorial computations to get the result. Chapter 5 presents the technique of Ottaviani and Rubei [OR06] for computing