quaderni di matematica

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1 quaderni di matematica volume 21 edited by Dipartimento di Matematica Seconda Università di Napoli Published with the support of Seconda Università di Napoli

2 quaderni di matematica Published volumes 1 - Classical Problems in Mechanics (R. Russo ed.) 2 - Recent Developments in Partial Differential Equations (V. A. Solonnikov ed.) 3 - Recent Progress in Function Spaces (G. Di Maio and Ľ. Holá eds.) 4 - Advances in Fluid Dynamics (P. Maremonti ed.) 5 - Methods of Discrete Mathematics (S. Löwe, F. Mazzocca, N. Melone and U. Ott eds.) 6 - Connections between Model Theory and Algebraic and Analytic Geometry (A. Macintyre ed.) 7 - Homage to Gaetano Fichera (A. Cialdea ed.) 8 - Topics in Infinite Groups (M. Curzio and F. de Giovanni eds.) 9 - Selected Topics in Cauchy-Riemann Geometry (S. Dragomir ed.) 10 - Topics in Mathematical Fluids Mechanics (G.P. Galdi and R. Rannacher eds.) 11 - Model Theory and Applications (L. Belair, Z. Chatzidakis et al. eds.) 12 - Topics in Diagram Geometry (A. Pasini ed.) 13 - Complexity of Computations and Proofs (J. Krajíček ed.) 14 - Calculus of Variations: Topics from the Mathematical Heritage of E. De Giorgi (D. Pallara ed.) 15 - Dispersive Nonlinear Problems in Mathematical Physics (P. D Ancona and V. Georgev eds.) 16 - Kinetic Methods for Nonconservative and Reacting Systems (G. Toscani ed.) 17 - Set Theory: Recent Trends and Applications (A. Andretta ed.) 18 - Selection Principles and Covering Properties in Topology (Lj.D.R. Kočinac ed.) 20 - Mathematical Modelling of Bodies with Complicated Bulk and Boundary Behavior (M. Šilhavý ed.) 21 - Vector Bundles and Low Codimensional Subvarieties: State of the Art and Recent Developments (G. Casnati, F. Catanese, and R. Notari eds.) Next issues Trends in Incidence and Galois Geometries: a Tribute to Giuseppe Tallini (F. Mazzocca, N. Melone and D. Olanda eds.) Numerical Methods for Balance Laws (G. Puppo and G. Russo eds.) Theory and Applications of Proximity, Nearness and Uniformity (G. Di Maio and S. Naimpally eds.) Nonlinear Elliptic Problems: the Notion of Solution with Non Finite Energy (A. Alvino ed.)

3 Vector Bundles and Low Codimensional Subvarieties: State of the Art and Recent Developments edited by Gianfranco Casnati, Fabrizio Catanese, and Roberto Notari

4 Authors addresses: Andrea Bruno Department of Mathematics Università degli Studi Roma Tre Largo San Leonardo Murialdo, 1 I Rome, Italy bruno@mat.uniroma3.it Gianfranco Casnati Department of Mathematics Politecnico di Torino Corso Duca degli Abruzzi, 24 I Torino, Italy casnati@calvino.polito.it Luca Chiantini Dipartimento di Scienze Matematiche e Informatiche Università degli Studi di Siena Pian dei Mantellini, 44 I Siena, Italy chiantini@unisi.it Philippe Ellia Department of Mathematics Università degli Studi di Ferrara Via Machiavelli, 35 I Ferrara, Italy phe@dns.unife.it Claudio Fontanari Department of Mathematics Politecnico di Torino Corso Duca degli Abruzzi, 24 I Torino, Italy claudio.fontanari@polito.it Ugo Bruzzo Scuola Internazionale Superiore di Studi Avanzati Istituto Nazionale di Fisica Nucleare Sezione di Trieste Via Beirut, 2-4 I Trieste, Italy; bruzzo@sissa.it Fabrizio Catanese Lehrstuhl Mathematik VIII Mathematisches Institut der Universität Bayreuth, NW II D Bayreuth, Deutschland fabrizio.catanese@uni-bayreuth.de Laura Costa Facultat de Matemàtiques Departament d Algebra i Geometria Gran Via de les Corts Catalanes, 585 E Barcelona, Spain costa@ub.edu Daniele Faenzi Department of Mathematics U. Dini Università di Firenze Viale Morgagni, 67/a I Florence, Italy faenzi@math.unifi.it Oscar García-Prada Instituto de Ciencias Matemáticas CSIC, Serrano 121 E Madrid, Spain oscar.garcia-prada@uam.es

5 Beatriz Graña Otero Departamento de Matemáticas and Instituto de Física Fundamental y Matemáticas Universidad de Salamanca Plaza de la Merced, 1-4 E Salamanca, Spain beagra@usal.es Roberto Notari Dipartimento di Matematica Politecnico di Milano Via Bonardi, 9 I Milano, Italy roberto.notari@polimi.it Luca Scala Max-Planck-Institut für Mathematik Vivatsgasse, 7 D Bonn, Germany scala@mpim-bonn.mpg.de Fabio Tonoli Department of Mathematics University of Trento Via Sommarive, 14 I Trento, Italy tonoli@science.unitn.it Rosa Maria Miró-Roig Department of Algebra and Geometry Faculty of Mathematics University of Barcelona Gran Via de les Corts Catalanes, 585 E Barcelona, Spain miro@mat.ub.es Giorgio Ottaviani Department of Mathematics U. Dini Università di Firenze Viale Morgagni, 67/a I Florence, Italy ottavian@math.unifi.it José Carlos Sierra Departamento de Álgebra Facultad de Ciencias Matemáticas Pza. de las Ciencias, 3 Universidad Complutense de Madrid E Madrid, Spain jcsierra@mat.ucm.es Alessandro Verra Department of Mathematics Università degli Studi Roma Tre Largo San Leonardo Murialdo, 1 I Rome, Italy verra@mat.uniroma3.it

6 ISBN

7 Contents Preface Geometry of Effective Divisors on Moduli Spaces 1 Andrea Bruno, Claudio Fontanari and Alessandro Verra Numerical Properties of Higgs Bundles 13 Ugo Bruzzo and Beatriz Graña Otero A Remarkable Moduli Space of Rank 6 Vector Bundles Related to Cubic Surfaces 41 Fabrizio Catanese and Fabio Tonoli On a Class of Surfaces in P 4 89 Luca Chiantini Low Rank Vector Bundles on the Blow Up Bl s (P n ) 109 Laura Costa Codimension Two Subvarieties and Related Questions 129 Philippe Ellia A Remark on Pfaffian Surfaces and acm Bundles 209 Daniele Faenzi Involutions of the Moduli Space of SL(n, C)-Higgs Bundles and Real Forms 219 Oscar García-Prada Lectures on Moduli Spaces of Vector Bundles on Algebraic Varieties 239 Rosa Maria Miró-Roig Symplectic Bundles on the Plane, Secant Varieties and Lüroth Quartics Revisited 315 Giorgio Ottaviani McKay Correspondence, Hilbert Schemes and Strange Duality 353 Luca Scala Low Codimensional Subvarieties of Projective Spaces Containing a Family of Very Degenerate Divisors 375 José Carlos Sierra

8 Mathematics subject classification Geometry of Effective Divisors on Moduli Spaces: 14H10, 14K10. Numerical Properties of Higgs Bundles: 14F05, 32L05, 32L20. A Remarkable Moduli Space of... : 14J60, 14F05, 14D20, 16E05, 14J25. On a Class of Surfaces in P 4 : 14J25, 14M07. Low Rank Vector Bundles on the Blow Up Bl s(p n ): 14F05. Codimension Two Subvarieties and Related Questions: 14M07, 14M10, 14C20. A Remark on Pfaffian Surfaces and acm Bundles: 14J60, 13C14. Involutions of the Moduli Space of... : 14H60, 57R57, 58D29. Lectures on Moduli Spaces of Vector Bundles on Algebraic Varieties: 14F05. Symplectic Bundles on the Plane,... : 14J60, 14F05, 14H50, 14N15, 15A72. McKay Correspondence, Hilbert Schemes and Strange Duality: 14C05, 14F05. Low Codimensional Subvarieties of Projective Spaces... : 14M07, 14N05, 14D06.

9 Preface This volume contains the Proceedings of the School and Workshop on Vector Bundles and Low Codimensional Varieties held in Povo-Trento (Italy) from September 11 to September 16, The meeting was jointly supported by the CIRM-ITC (at present CIRM-Fondazione Bruno Kessler), by the Dipartimento di Matematica del Politecnico di Torino and by the Dipartimento di Matematica della Università di Torino (in the realm of the joint project Geometria sulle Varietà Algebriche cofinanced by the MIUR, the Italian Ministry for University and Research). The volume contains twelve papers related to the topics of the meeting. Some of the papers are Lecture Notes of the school, some are expository, some other are purely research papers. All the papers have been refereed by peer reviewers. The papers by Ph. Ellia and R.M. Miró-Roig are essentially Lecture Notes, covering in detail the contents of the respective lectures, which were delivered during the School. They both amply illustrate the different aspects and facets of vector bundle theory. While the lectures by R. M. Miró-Roig deal with moduli spaces of (semi)stable vector bundles on smooth surfaces and on higher dimensional smooth varieties, the lectures of Ph. Ellia concentrate on the problem of existence and properties of sections of vectors bundles of low rank (essentially of rank two) on a projective space, with a particular interest in Hartshorne s famous conjecture on smooth subvarieties of codimension two in projective space. The other papers by A. Bruno and C. Fontanari and A. Verra, U. Bruzzo and B. Graña-Otero, L. Chiantini, F. Catanese and F. Tonoli, L. Costa, D. Faenzi, O. García-Prada, G. Ottaviani, L. Scala, J.C. Sierra give contributions to some different aspects of the two main themes of the meeting. Some of them are related to the talks delivered by the authors during the Workshop. The diversity of these contributions reflects, without any pretention of completeness,

10 the wide range of investigations currently dealing with these central subjects relating classical and modern algebraic geometry. The theory of sheaves and vector bundles arose naturally in the attempt to infer global properties of objects whose local behaviour at each point of a certain ambient space is quite easy to understand, the typical example being the De Rham theory of closed and exact differential forms (every closed form is locally exact, but the De Rham cohomology measures the obstruction to the converse through the integrals over closed submanifolds). The original notion of vector bundle was first introduced in the context of algebraic topology: the first complete and standard reference is the beautiful book by N. Steenrod The topology of fibre bundles, published in The first natural examples arising in topology and geometry are the Grassmann and flag varieties parametrizing linear subspaces of a fixed dimension inside a vector, affine or projective space. These lead to the theory of universal bundles and classifying spaces, a main theme of modern algebraic topology. Natural objects in differential topology are the tangent and the normal bundle to a differentiable submanifold of a given manifold, together with their associated bundles (bundles associated to any representation of the linear group, such as the exterior and the symmetric powers) and their characteristic classes: Stiefel- Whitney classes, Pontrjagin and Chern classes. Pretty soon the theory of vector bundles was related to the theory of coherent sheaves and their cohomology (in this language, to a bundle corresponds the sheaf of local holomorphic sections, a so called locally free sheaf). The notion of vector bundles had as its precursor in algebraic geometry the study of ruled varieties and of scrolls, and the first classification results, obtained by Corrado Segre at the end of XIX century, were later rediscovered at the crucial historical moment in the fifties of the past century when the theory of vector bundles revolutionized algebraic geometry thanks to the work of Kodaira, Nakano, Serre, Atiyah, Grothendieck and others. In particular we owe to A. Grothendieck the classification of bundles on the complex projective line, and to M. Atiyah the classification of vector bundles over elliptic curves. Both Grothendieck s and Atiyah s results are heavily relying on deep and subtle cohomological methods which revealed themselves to be widely helpful and

11 which rely on previous work of S.S. Chern and others on characteristic classes. In particular, the results by Grothendieck and Atiyah opened the way to natural question concerning the investigation of families of vector bundles over a base X other than a curve of genus 0 or 1. In doing this, quite naturally makes its appearance the notion of stability together with its relation to G.I.T. (Geometric Invariant Theory); precisely when one attempts to construct an algebraic variety parametrizing isomorphism classes of bundles, over a fixed base variety X, having given Chern classes. This variety, when it exists, is called the moduli space. Quite often a moduli space exists only when one restricts the consideration only to these bundles which are stable or semistable (there are here subtle questions, indeed there are several notions of stability, as for instance Mumford-Takemoto s slope stability, and Gieseker s and Maruyama s Hilbert polynomial stability: and according to the different applications, one notion may turn out to be more appropriate than the other). After that M. Maruyama proved the existence of a coarse moduli space for stable algebraic vector bundles of rank r with fixed Chern classes on any projective manifold X, many efforts were spent in order to study these moduli spaces over a fixed variety X. This task turned out to be much more complicated and difficult than foreseen, and nowadays much of what is known tends to concentrate to the case on rank 2 bundles, which are related to codimension 2 subvarieties via the Schwarzenberger-Serre construction. The Lecture Notes by R.M. Miró-Roig, which as we said cover the topics of the lectures she delivered during the School, illustrate carefully the moduli problem for vector bundles, the notions of stability and semistability, and the construction technique introduced by G. Horrocks and called Monads construction. This technique, which was later amply generalized by Beilinson, turns out to be very useful for the description of many families of bundles, and provides many interesting results, hints and suggestions regarding the structure of moduli spaces of vector budles on varieties of dimension at least three. Moduli spaces of vector bundles, possibly with particular properties, are not only interesting by their own but often play an important role in the construction of varieties with some fixed properties. For instance F. Catanese and F. Tonoli in their research paper investigate carefully the moduli space of

12 simple vector bundles of rank 6 ( simple means that there are no non-scalar automorphisms) on projective 3-space which have fixed Chern classes 1, 3, 6 and 4. They describe in detail the irreducible component of this moduli space corresponding to vector bundles with minimal cohomology, and interpret such locus as a quotient of the variety of suitable 3-tensors. As the authors explain in the first section of the paper, these bundle are particularly interesting since they are strictly related to the construction of surfaces of degree 6 in projective 3-space endowed with an even set of 56 nodes. Another example of such an intimate relationship between moduli spaces of vector bundles and varieties with some fixed properties can be found in the paper by G. Ottaviani. The starting point is the remark that the equation of a sufficiently high secant variety to a suitable defective Segre embedding, the socalled Strassen equation, is almost identical to an equation naturaly associated by W. Barth to a moduli space of vector bundles of rank 2 and with even first Chern class over the projective plane. Then the author tries to explore further this interesting connection employing, in his analysis, other techniques and objects from multilinear algebra, classical plane curve theory, and the theory of higher rank (symplectic) vector bundles. We already mentioned the fundamental contributions to the theory of algebraic vector bundles due to J.P. Serre and R.L.E. Schwarzenberger: they explained how to construct vector bundles as extensions of sheaves related to certain subvarieties. Such a construction generalizes the well-known correspondence on a variety between line bundles and Cartier divisors (i.e., subvarieties locally defined by the vanishing of a single regular function). The study of some particular Cartier divisors on a fixed variety and of their effectiveness is strictly related with its birational geometry. E.g. J. Harris and D. Morrison made some conjectures concerning the socalled slope of the moduli space of curves of genus g: this notion turns out to be crucial in order to extend to all genera the results by Mumford, Harris and Eisenbud on the Kodaira dimension of the moduli space of curves of genus g 24. The paper by A. Bruno, C. Fontanari, A. Verra is framed in this perspective. There the authors compute the slope of effective divisors on the moduli spaces of curves of genera 13 and 14 and of abelian varieties of genus 5.

13 Passing to another topic, a very degenerate divisor on a smooth n-dimensional variety X embedded in the r-dimensional projective space is by definition an effective divisor spanning a linear subspace of dimension at most n. It is a classical problem to classify varieties containing an algebraic family of such very degenerate divisors. In the expository note by J.C. Sierra are collected some new and old results on such kind of varieties, expecially in the most interesting cases r = 2n 1 and r = 2n, presenting some related conjectures and open problems as well. The Schwarzenberger-Serre approach became quickly the standard way to deal with vector bundles of low rank. In this context R. Hartshorne enunciated his fundamental conjecture on subvarieties of a projective space in 1973: the simplest case of his conjectures is the statement that in a projective space of dimension at least 6 any smooth subvariety of codimension two is a complete intersection of two hypersurfaces. This conjecture, in spite of the many efforts, is still unproven, together with many technical intermediate steps (except the 1-normality conjecture proven by F. Zak). The Lecture Notes by Ph. Ellia give a complete picture of such contributions to the above theme, suggesting some possible approaches and many related ideas. Also this paper is a faithful account of the lectures delivered by the author during the School. The weakened Hartshorne conjecture states the non existence of indecomposable rank two vector bundles on n-dimensional projective space when n 6. Thus it is intrinsically interesting to understand if it is possible to construct such kind of indecomposable bundles on a variety of high dimension birationally related to projective spaces. L. Costa in her research paper deals with low rank vector bundles on the blow up of n-dimensional projective space in a finite set consisting of at most n + 1 points, constructing some families of stable, hence indecomposable, rank two vector bundles. The problem of finding easy criteria for deciding whether a vector bundle is decomposable is old and, in general, open. The first answers are given by Grothendieck and Atiyah as a by-product of their above mentioned contributions. In general, answers are not evident without the help of more refined homological techniques. The first very important contribution is due to G. Hor-

14 rocks, who classified vector bundles on projective spaces of dimension n 2 up to stable equivalence, proving in particular the well-known splitting criterion asserting that the vector bundles which split as a direct sum of line bundles (these are the vector bundles of rank 1) are exactly the acm (arithmetically Cohen Macaulay) bundles, i.e., vector bundles with vanishing intermediate cohomology. Such a splitting criterion does not hold in the same form over other varieties. In particular, there exist indecomposable acm bundles and thus it is quite natural to investigate such a property more deeply; many papers have been published after the one by G. Horrocks, dealing with acm bundles over particular classes of varieties, such as quadrics, some other Fano varieties, Grassmannians and hypersurfaces of degree d in the n-dimensional projective space with n 4. When n = 3 the classification is known only when d 5. For higher values of d only few results are known and the short note by D. Faenzi gives a contribution in this case. The method used there is to construct surfaces as degeneracy loci of suitable morphisms between vector bundles. Such an approach is used also in the paper by L. Chiantini in order to propose some problems regarding subcanonical surfaces in the 4-dimensional projective space. Vector bundles and the related notions deeply permeated in the last 50 years different areas of modern research, both in mathematics and in physics, and also in other fields. In fact, bundles were the mathematical tool to a proper treatment of gauge theories in physics. This mathematization brought to a strong interest of the mathematical community in the theory of instanton bundles. Through the Atiyah-Drinfeld-Hitchin-Manin construction and the correspondences found by Ward and Penrose, the mathematical counterpart of the theory of instanton bundles relates to the existence of certain algebraic vector bundles on projective 3-space. The theory of Yang-Mills connections, and the selfduality and antiselfduality equations, have been the inspiration and source of the Donaldson and Seiberg-Witten theories, which provided invariants of differentiable structures on topological manifolds, leading in particular to amazing discoveries concerning the topology and differential topology of four-manifolds. Another important issue where vector bundles over curves appear to play a

15 very important role is the Langlands program, connecting number theory and the representation theory of certain algebraic groups. Its geometric reformulation relates, in simple cases, the l-adic representations of the étale fundamental group of an algebraic curve (defined over a finite field) to elements of the derived category of vector bundles on the curve. The contributions to the present volume by O. García-Prada and the one by U. Bruzzo and B. Graña-Otero, concerning the theory of Higgs bundles (bundles endowed with a Higgs field, i.e., an endomorphism with values in the sheaf of differentials, satisfying an integrability condition) achieve this goal. In particular in the first of the above papers the author shows how the moduli space of representations of the fundamental group of a compact Riemann surface X in any real form of SL(n, C) can be identified with the fixed point subvariety of a certain involution of the moduli space of SL(n, C)-Higgs bundles on X. In the second one, the authors confront the notions of numerically effectiveness and semistability of Higgs bundles over Kähler manifolds and over complex projective manifolds. Finally, L. Scala in his paper surveys recent developments concerning the strange duality conjecture of Le Potier. This conjecture concerns pairs of moduli spaces of vector bundles over an algebraic surface, whose Grothendieck classes are orthogonal, and asserts that a certain morphism is indeed an isomorphism. The author shows how the MacKay correspondence, relating G- equivariant sheaves on a G-variety with sheaves over the fixed locus of the punctual Hilbert scheme, may be used to contribute some applications towards the proof of the conjecture in special cases. We heartily thank all the authors for kindly agreeing to our proposal, thus contributing their articles to the present volume, the referees for their anonymous but essential collaboration and for their generous effort, the participants in the school and Workshop for their enthusiasm, and finally Prof. P. Maremonti, the editor of the Quaderni di Matematica, for suggesting the possibility to edit this volume and helping us to realize this goal. Gianfranco Casnati, Fabrizio Catanese, and Roberto Notari