MODULI OF BRIDGELAND STABLE OBJECTS ON AN ENRIQUES SURFACE BY HOWARD J. NUER A dissertation submitted to the Graduate School|New Brunswick Rutgers, The State University of New Jersey In partial fulfillment of the requirements For the degree of Doctor of Philosophy Graduate Program in Mathematics Written under the direction of Lev Borisov And approved by New Brunswick, New Jersey May, 2016 ABSTRACT OF THE DISSERTATION Moduli of Bridgeland stable objects on an Enriques surface By HOWARD J. NUER Dissertation Director: Lev Borisov We construct projective moduli spaces of semistable objects on an Enriques surface for generic Bridgeland stability condition. On the way, we prove the non-emptiness of s MH;Y (v), the moduli space of Gieseker stable sheaves on an Enriques surface Y with Mukai vector v of positive rank with respect to a generic polarization H. In the case of a primitive Mukai vector on an unnodal Enriques surface, i.e. one containing no smooth rational curves, we prove irreducibility of MH;Y (v) as well. Using Bayer and Macr`ı’s construction of a natural nef divisor associated to a stability condition, we explore the relation between wall-crossing in the stability manifold and the minimal model program for Bridgeland moduli spaces. We give three applications of our machinery to obtain new information about the classical moduli spaces of Gieseker-stable sheaves: 1) We obtain a region in the ample cone of the moduli space of Gieseker-stable sheaves over Enriques surfaces. 2) We determine the nef cone of the Hilbert scheme of n points on an unnodal Enriques surface in terms of its half-pencils and the Cossec-Dolgachev φ-function. 3) We recover some classical results on linear systems on unnodal Enriques surfaces and obtain some new ones about n-very ample line bundles. ii Acknowledgements First and foremost, I must express gratitude to Hashem. While the hard work going into this dissertation are mine to be proud of, the mathematical talent and inspiration that made this endeavor successful were gifts from Him. I am also deeply indebted to my advisor, Lev Borisov, for his constant generous support and guidance. He encouraged me to begin doing research very early on in my graduate school career, and I owe a lot of my success to this. He taught me to learn new subjects by doing research in those subjects, which is likely responsible for the number and breadth of topics touched in my papers. Lev initiated our research on a joint project, but encouraged me to develop my own interests and projects. Even though much of my later research focused on areas that were not of particular interest to him, Lev's amazing insight and intuition, coupled with his constant willingness to discuss my research and act as a sounding board proved invaluable to my success. This project was first inspired by the talks of Arend Bayer and Emanuele Macr`ıat the Graduate Student Workshop on Moduli Spaces and Bridgeland Stability at UIC, and I would like to thank Arend and Emanuele for their encouragement of this undertaking as well as for very helpful discussions. I would also like to thank the numerous other mathematicians around the world who provided encouragement, insight, and helpful discussions throughout this project, including D. Abramovich, T. Bridgeland, K. Yosh- ioka, I. Coskun, I. Dolgachev, A. Maciocia, E. Markman, H. Kim, M. Thaddeus, and G. Sacca. I owe a great deal to my local support system here at Rutgers, including Professors Steve Miller, Chris Woodward, Chuck Weibel, and Anders Buch. I would especially like to thank my close friends in the department who listened to my stream-of-consciousness diversions in the office, whether they be about math or just life. Most of all, I thank iii my former office-mates, Sjuvon, Ed, Charles, Knight, Pat, and Zhuohui. Thank you for listening to my craziness and know that talking math with you guys taught me more than any book I read on my own. I would also like to thank Lev's former student Zhan Li for many hours of learning new subjects together. Finally, I am happy to thank my loving family. I thank Miriam, the love of my life and my devoted wife, for listening to my fears and insecurities and supporting me through them. I would like to thank her in addition for working so hard in the hospital and in the clinic to support our growing family while I pursue my dream. I want to thank my 3 beautiful children, Shalom, Gabi, and Shaina, for forcing me to both manage my time well and escape the black hole that math research can become. You may have been the true secret to my success. Lastly, I'd like to thank my mom who always encouraged me to do whatever made me happy and taught me the value of independence. When I was a kid, you always asked me if I had done my best. This stuck with me and gave rise to my internal motivation and drive. iv Dedication To my math teachers, David Doster and Michael Lutz, for opening the door for me to a whole new world. v Table of Contents Abstract :::::::::::::::::::::::::::::::::::::::: ii Acknowledgements ::::::::::::::::::::::::::::::::: iii Dedication ::::::::::::::::::::::::::::::::::::::: v 1. Notation and Conventions ::::::::::::::::::::::::::: 1 2. Introduction ::::::::::::::::::::::::::::::::::: 3 2.1. Overview...................................3 2.2. Previous results................................6 2.3. An Outline and Summary of Main Results.................7 2.4. Open questions................................ 13 3. Review: Enriques surfaces ::::::::::::::::::::::::::: 15 3.1. First properties................................ 15 3.2. Divisors on Enriques surfaces........................ 16 3.3. The algebraic Mukai lattice......................... 17 4. Review: Moduli spaces of semistable sheaves :::::::::::::: 21 4.1. Slope stability................................. 21 4.2. Gieseker stability............................... 22 4.3. Moduli spaces of semistable sheaves.................... 23 4.3.1. Moduli spaces for Gieseker semistability.............. 23 4.3.2. Moduli spaces for µ-semistability.................. 23 4.4. Wall-and-chamber structure on Amp(X).................. 24 4.4.1. Suitable polarizations........................ 24 4.5. (Quasi-)universal families.......................... 25 vi 4.6. Moduli of sheaves on Enriques surfaces: What is known?......... 26 5. Review: Bridgeland stability ::::::::::::::::::::::::: 30 5.1. Bridgeland stability conditions....................... 30 5.2. Stability conditions on K3 and Enriques surfaces............. 32 5.2.1. Space of stability conditions for a K3 surface........... 32 5.2.2. Space of stability conditions for an Enriques surface via induction 33 5.3. The Wall-and-Chamber structure...................... 34 6. Moduli stacks of semistable objects ::::::::::::::::::::: 38 6.1. Basic properties of semistable objects................... 38 6.2. Moduli stacks of Bridgeland semistable objects.............. 41 6.3. The Geometry of the Morphism π∗ ..................... 46 6.4. Singularities of Bridgeland moduli spaces and their canonoical divisor. 49 7. Interlude: New results on moduli of stable sheaves on Enriques sur- faces :::::::::::::::::::::::::::::::::::::::::: 51 7.1. Classification of chern classes........................ 51 7.2. A quick proof of non-emptiness and irreducibility using Bridgeland sta- bility and derived category techniques................... 54 7.3. The existence of stable sheaves in the non-primitive case......... 57 7.4. Extending to nodal Enriques surfaces via deformation theory...... 60 8. Projectivity of Coarse Moduli Spaces ::::::::::::::::::: 61 8.1. Non-emptiness................................ 61 8.2. The unnodal case............................... 62 8.3. The nodal case................................ 68 9. Bridgeland wall-crossing and birational geometry :::::::::::: 71 9.1. A Natural Nef Divisor............................ 71 9.2. Flops via Wall-Crossing........................... 73 vii 10.Applications ::::::::::::::::::::::::::::::::::: 77 10.1. Moduli of stable sheaves........................... 77 10.2. Hilbert Schemes of points on an Enriques Surface............. 81 10.3. Applications to linear systems........................ 87 11.Appendix: Some neo-classical proofs of non-emptiness and irreducibil- ity for moduli of sheaves :::::::::::::::::::::::::::::: 92 11.1. Non-emptiness and irreducibility of moduli spaces in rank 4 when c1:f = ±1........................................ 92 11.2. Appendix: Dimension estimates for Brill-Noether loci on Hilbert schemes of points.................................... 115 References ::::::::::::::::::::::::::::::::::::::: 117 viii 1 Chapter 1 Notation and Conventions Throughout, we work over C, and X will denote a smooth projective variety over C unless otherwise specified. For a (locally-noetherian) scheme (or algebraic space) S,Db(S) denotes the bounded derived category of coherent sheaves, Dqc(S) the unbounded derived category of quasi- coherent sheaves, and DS-perf (S×X) the category of S-perfect complexes. (An S-perfect complex is a complex of OS×X -modules which locally, over S, is quasi-isomorphic to a bounded complex of coherent shaves which are flat over S.) We will abuse notation and denote all derived functors as if they were underived. We denote by pS and pX the two projections from S × X to S and X, respectively. Given E 2 Dqc(S × X), we denote the Fourier-Mukai functor associated to E by ∗ ΦE ( ) := (pX )∗ (E ⊗ pS( )) : We define the numerical Grothendieck group of a triangulated category T by Knum(T ) := K(T )= Ker(χ), where K(T ) is