Durham E-Theses Extension maps and the moduli spaces of rank 2 vector bundles over an algebraic curve Gronow, Michael Justin How to cite: Gronow, Michael Justin (1997) Extension maps and the moduli spaces of rank 2 vector bundles over an algebraic curve, Durham theses, Durham University. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/5081/ Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in Durham E-Theses • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full Durham E-Theses policy for further details. Academic Support Oce, Durham University, University Oce, Old Elvet, Durham DH1 3HP e-mail: [email protected] Tel: +44 0191 334 6107 http://etheses.dur.ac.uk 2 The copyright of this thesis rests with the author. No quotation from it should be pubHshed without the written consent of the author and information derived from it should be acknowledged. Extension Maps and the Moduli Spaces of Rank 2 Vector Bundles over an Algebraic Curve by Michael Justin Gronow A thesis presented for the degree of Doctor of Philosophy June 1997 Department of Mathematical Sciences, University of Durham, South Road, Durham, DHl 3LE, England 0 NOV 199? Abstract. Extension Maps and the Moduli Spaces of Rank 2 Vector Bundles over an Algebraic Curve Michael Justin Gronow Let <5Wc(2,A) be the moduli space of rank 2 vector bundles with determinant A on an algebraic curve C. This thesis investigates the properties of a rational map ^Ud,A SUc(2, A) where FUd,A is a projective bundle of extensions over the Jacobian J'^{C). In doing so the degree of the moduli space SUc{^, Oc) is calculated for non- hyperelliptic curves of genus four (3.4.2). Information about trisecants to the Kummer variety JC C SUc{'^,Oc) is obtained in sections 4.3 and 4.4. These sections describe the varieties swept out by these trisecants in the fibres of IPJ7i,0c ~^ ^^i^) curves of genus 3, 4 and 5. The fibres of over E G <S^c(2, A) are then studied. For certain values of d these correspond to the family of maximal line subbundles of E. These are either zero or one dimensional and a complete description of when these famihes are smooth is given (5.4.9), (5.4.10). In the one dimensional case its genus is also calculated (if connected) (5.5.5). Finally a correspondence on the curve fibres is shown to exist (5.6.2) and its degree is calculated (5.6.5). This in turn gives some information about multisecants to projective curves (5.7.4), (5.7.7). Preface. Extension Maps and the Moduli Spaces of Rank 2 Vector Bundles over an Algebraic Curve Michael Justin Gronow This work has been sponsored by the U.K. Engineering and Physical Sciences Re• search Council. The thesis is based on research carried out between October 1993 and June 1997 under the supervision of Dr W.M.Oxbury. It has not been submitted for any other degree either at Durham or at any other University. No claim of originality is made for the material presented in chapter 1 or chapter 2 apart from (2.2.8). The material in chapters 3, 4 and 5 is original apart from the introductory remarks at the beginning of each chapter, sections 3.1, 3.3 and 4.1 which review the main results of [NR2], [BNR] and [OPP] that will be needed in this thesis and sections 4.2 and 5.1 which give a short account of elementary results on ruled surfaces and determinantal varieties respectively. It has also been necessary to include other authors results throughout these chapters and it is clearly stated when this occurs. Copyright © 1997 by Michael J. Gronow The copyright of this thesis rests with the author. No quotation from it should be published without the author's prior written consent and information derived from it should be acknowledged. 3 Acknowledgements. First and foremost I would like to thank my supervisor Bill Oxbury for many helpful suggestions and conversations throughout my time at Durham. I would also like to express my gratitude to my friends and colleagues in the Math• ematics Department for making my time in Durham all the more enjoyable. Special thanks must go to Helen Fawley and Karen McGaul for entertaining lunchtimes and getting me addicted to cryptic crosswords, Michael Young and Mansour Aghasi for their technical advice and Ahstair Maclntyre for his enviable quahty of being able to cheer me up through the hard times. Last but by no means least I would like to thank my parents for their support and encouragement. This thesis is dedicated to them. Contents Chapter 1. Introduction 7 1.1. Moduli spaces of vector bundles 9 1.2. Stratification of the moduli space 13 1.3. Extension maps 14 1.4. Cohomology and the Verlinde formulae 15 Chapter 2. The fibrewise extension map 18 2.1. Moduli spaces of pairs 22 2.2. The genus 2, odd degree moduli space 25 Chapter 3. The g-plane ruling 31 3.1. The genus 3, even degree moduli space 32 3.2. Degree of g-plane ruling 33 3.3. Spectral Curves 40 CONTENTS 3.4. The class of the g-plane ruling 43 Chapter 4. Trisecants to the Kummer variety 46 4.1. Fay trisecants 46 4.2. Ruled Surfaces 50 4.3. Quadrisecants to hyperelliptic Kummers 51 4.4. Trisecants to non-hyperelliptic Kummers 55 Chapter 5. Maximal line subbundles 72 5.1. Determinantal Varieties 73 5.2. Preliminary results on maximal subbundles 74 5.3. Fibres of the extension map 77 5.4. Smoothness of the fibres 80 5.5. Degree of canonical line bundle on curve fibres 88 5.6. Correspondence on curve fibres 94 5.7. Multisecants to projective curves 101 5.8. Connectedness of fibres 106 Bibliography 109 CHAPTER 1 Introduction This thesis studies the moduli spaces of rank 2 bundles with fixed determinant bundle A from the viewpoint of extensions or short exact sequences. In particular there exists a projective bundle of extensions FUd,A over the Jacobian of degree d line bundles on C and a rational map to the moduli space: It is this map that will be the main object of study throughout and the thesis is structured in the following way: The rest of this chapter gives an introduction to moduh spaces of vector bundles and reviews some well known results. The projective bundle of extensions and the map are then formally defined and in the last section the rational cohomology of SUc{2, A) and the Verlinde formulae for rank 2 bundles is discussed. Chapter 2 looks at restricted to a single fibre of VUd,A —^ J'^{C) and reviews work of Bertram [Be] and Thaddeus [Th] done in this area. The final part of the chapter shows how Thaddeus' "flips" construction is equivalent, in the genus 2 case, to a classical construction showing the rationality of the quadratic line complex (2.2.6), (2.2.7) and (2.2.8). Chapters 3 and 4 study ta over the whole of TUd,A but restricts to the special case 7 1. INTRODUCTION d = 1, A = Oc- ^1 then has the particularly nice property that it is linear and injective on the fibres of PC/i,Oc J^iQ with respect to a line bundle C on <S^/c(2, Oc) (2.0.4). This then allows one to calculate the degree of the image of ti (3.2.9) (with respect to £) and, in the case when g = 4:, the degree of the moduli space itself (3.4.2). It has also been shown [OPP] that the trisecants to the Kummer variety /C C SUc{2, Oc) are also trisecants to the curve C in some fibre of fUi^Oc ~^ J^i^)- Chapter 4 describes the variety of trisecants in each of these fibres for curves of genus 3, 4 and 5. The final chapter is concerned with the fibres of ej, over a bundle E € SUc{2,A). For certain values of d these correspond to the maximal hne subbundles of E and are either zero or one dimensional. (5.4.9) and (5.4.10) describes exactly when these fibres are smooth. In the smooth one dimensional case the degree of its canonical line bundle is calculated (5.5.3) (and hence its genus if the fibre is connected.) A correspondence is then shown to exist (5.6.2) on the curve fibres and its degree is calculated (5.6.5). It will then be shown how this can lead to information about certain multisecants to projective curves (5.7.4), (5.7.7). Finally a list of sufficient conditions (5.8.1) for the one dimensional fibres to be connected is given. Unless otherwise stated the following notation will be used throughout: C - a non-hyperelliptic compact Riemann surface of genus g > 2. S^C - the z'-th symmetric power of C. E,F - rank two holomorphic vector bundles on C. ' A - determinant line bundle. In particular A = Oc or Oc{p) (for some p G C) depending on the degree of the rank 2 bundle.