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A tale of two moduli spaces: Hilbert schemes of singular curves and moduli of elliptic surfaces by Dori Bejleri B. Sc., California Institute of Technology; Pasadena, CA 2013 M. Sc., Brown University; Providence, RI, 2016 A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics at Brown University Providence, Rhode Island May 2018 © Copyright 2018 by Dori Bejleri This dissertation by Dori Bejleri is accepted in its present form by the Department of Mathematics as satisfying the dissertation requirement for the degree of Doctor of Philosophy. Date Dan Abramovich, Ph. D., Advisor Recommended to the Graduate Council Date Brendan Hassett, Ph. D., Reader Date Melody Chan, Ph. D., Reader Approved by the Graduate Council Date Andrew G. Campbell Dean of the Graduate School iii Curriculum Vitae Dori Bejleri grew up in Connecticut and graduated salutatorian from Wolcott High School in 2009. He earned a Bachelor of Science degree in mathematics from the California Institute of Technology in 2013 and a Master of Science degree in mathematics from Brown University in 2016. He completed his PhD in algebraic geometry at Brown University in May 2018 under the supervision of Dan Abramovich. Publication list (1) K. Ascher and D. Bejleri, Stable pair compactifications of the moduli space of degree one del pezzo surfaces via elliptic fibrations, I. arXiv:1802.00805. (2) D. Bejleri, D. Ranganathan and R. Vakil, Motivic Hilbert zeta functions of curves are rational. arXiv:1710.04198. In revision at the Journal of the Institute of Mathematics of Jussieu. (3) D. Bejleri, The SYZ conjecture via homological mirror symmetry. arXiv:1710.05894. Contribution to the proceedings of the Superschool on derived categories and D-branes. (4) K. Ascher and D. Bejleri, Moduli of weighted stable elliptic surfaces and invari- ance of log plurigenera (with an appendix by G. Inchiostro). arXiv:1702.06107. iv (5) K. Ascher and D. Bejleri, Moduli of fibered surface pairs from twisted stable maps. arXiv:1612.00792. In revision at Mathematische Annalen. (6) K. Ascher and D. Bejleri, Log canonical models of elliptic surfaces. arXiv:1610.08033. Advances in Mathematics, Volume 320 (2017), 210-243. (7) D. Bejleri and D. Stapleton, The tangent space to the punctual Hilbert scheme. arXiv:1604.04915. Michigan Math Journal, Volume 66, Issue 3 (2017). (8) D. Bejleri and G. Zaimi, The topology of equivariant Hilbert schemes. arXiv:1512.05774. (9) D. Bejleri and M. Marcolli, Quantum field theory over F1. arXiv:1209.4837. Journal of Geometry and Physics, Volume 69 (2013). v Acknowledgements This thesis is the result of countless conversations with friends and colleagues, inspiration from mentors, and support from family. It wouldn’t be complete without thanking you. First I would like to thank my advisor Dan Abramovich. When I arrived at Brown to work with you, I could not have anticipated the amount of encouragement, support and guidance you would provide. Your enthusiasm for math is inspiring and your effect on my development as a mathematician cannot be understated. I would like to say thank you to my thesis committee, Brendan Hassett and Melody Chan, for your mentorship, feedback and many interesting conversations. I have been lucky to have had many other mentors throughout my career who helped me along this path. I am especially grateful to Anton Geraschenko, Tom Graber and Matilde Marcolli who early in my mathematical career introduced me to algebraic geometry and taught me how to do research, and to Davesh Maulik and Ravi Vakil whose guidance has been indispensable to my later mathematical endeavours. To Alex, Chris E., Chrispy, Gjergji, Josh and Wade: I discovered mathematics with you through countless late nights spent working on problem sets and enthusiastically vi discussing math way beyond our level. I am so grateful for your friendship and for giving me these memories which still color my feelings towards doing math. Mathematics is a social activity and wouldn’t have been half as fulfilling without many wonderful interactions with my collaborators and colleagues. Thank you Kenny Ascher, Dhruv Ranganathan, David Stapleton, and many others. To Audrey, Carol, Doreen, Larry, Lori and Sidalia, you made being at Brown such a welcoming and enjoyable experience. Thank you for everything you do for me and the other graduate students. During my time at Brown, I had the opportunity to interact with many wonderful algebraic geometers so I would like to thank Alicia Harper, Kuan-Wen Lai, June Park, Nathan Pflueger, Evangelos Routis, Martin Ulirsch, Zijian Yau, and Yuwei Zhu for creating such an enriching research community. What I will remember most from grad school isn’t the math I did, but rather the people I spent time with and the friendships we formed. Thank you to all of you for making this an unforgettable experience. Special thanks to Shamil Asgarli, Giovanni Inchiostro and Laura Walton for all the support you’ve given me and for keeping me sane. I would like to thank my family, especially my parents for your love and support and for sacrificing so much to put me on this path, and Grizzly for being a bright part of every day. Finally, to Jen, you are the love of my life and nothing I’ve accomplished would have been possible without you. I dedicate this thesis to you. vii Contents 1 Introduction 1 1.1 Hilbert schemes of points on singular curves . 2 1.2 Compact moduli spaces of elliptic surfaces . 6 1.2.1 An example . 14 1.2.2 Applications and further work . 15 1.2.3 Previous results . 17 I Hilbert schemes of points on singular curves 18 2 Preliminaries 19 2.1 The Grothendieck ring of varieties . 19 2.2 Hilbert schemes of points . 22 2.3 The geometry of singular curves . 23 2.3.1 The branch-length filtration and graded degenerations . 25 3 The Hilbert zeta function is rational 30 3.1 Reduction to a local calculation . 30 3.1.1 A stratification of the punctual Hilbert scheme . 32 viii 3.2 Degree–branch-length bounds . 34 3.3 Motivic stabilization for branch-length strata . 38 3.3.1 Conclusion of the proof of the Main Theorem . 41 3.4 Extended example: the coordinate axes . 42 3.4.1 The axes in three space . 45 3.5 Nonreduced curves . 46 3.5.1 Hilbert scheme of points on the plane . 47 3.5.2 Proof of Proposition 3.5.2 . 50 3.5.3 Locally planar uniformly thickened curves . 52 4 The Hilbert zeta function in families 54 4.1 Relative Hilbert schemes of points . 54 4.2 Singular curves and their deformations . 56 4.2.1 Equisingular families . 58 4.3 Proof of Theorem 1.1.2 . 62 II Compact moduli spaces of elliptic surfaces 64 5 Preliminaries 65 5.1 The minimal model program and moduli of stable pairs . 65 5.1.1 Semi-log canonical pairs . 65 5.1.2 Moduli spaces of stable pairs . 69 5.1.3 Vanishing theorems . 71 5.2 Elliptic surfaces . 74 5.2.1 Standard elliptic surfaces . 74 5.2.2 Singular fibers . 76 ix 6 Log canonical models of elliptic surfaces 78 6.1 Log canonical models of A-weighted elliptic surfaces . 78 6.1.1 Relative log canonical models . 79 6.1.2 Canonical bundle formula . 88 6.1.3 Pseudoelliptic contractions . 89 7 Moduli spaces of weighted stable elliptic surfaces 93 7.1 Weighted stable elliptic surfaces . 93 7.1.1 Moduli functor for elliptic surfaces . 95 7.1.2 Broken elliptic surfaces . 99 7.1.3 Formation of pseudoelliptic components . 103 7.2 Stable reduction . 105 7.2.1 Wall and chamber structure . 105 7.2.2 The birational transformations across each wall . 109 7.2.3 No other flipping contractions . 113 7.2.4 Explicit stable reduction . 114 8 Wall-crossing morphisms for Ev,A 119 8.1 Reduction morphisms . 119 8.1.1 Hassett’s moduli space . 119 8.1.2 Preliminary results . 120 8.1.3 Reduction morphisms . 121 8.2 Flipping walls . 124 8.2.1 The “wall” at a = 1 . 131 9 Invariance of log plurigenera for broken elliptic surfaces 134 x A Normalizations of algebraic stacks 158 Bibliography 164 xi List of Tables 5.1 Singular fibers of a smooth minimal elliptic surface . 77 6.1 Intersection pairings in a standard intermediate fiber . 86 xii List of Figures 1.1 An A-weighted broken elliptic surface. 13 1.2 The wall crossing transformations on the central fiber of a stable degeneration of a rational elliptic surface. 14 2,1,1,1 3.1 On the left, the stratum Hilb (X3, 0) whose closure contains three 1,1,1,1 zero dimensional strata corresponding to twisting Hilb (X3, 0) 3,1,1,1 along each of the three branches. On the right, the stratum Hilb (X3, 0) 3,2,1,1 whose closure contains coordinate lines inside of Hilb (X3, 0) and its permutations. 45 3 2 3.2 The reduced Hilbert scheme Hilb (X3, 0) consists of 4 copies of P glued along coordinate lines as depicted. The solid shaded strata 3,2,1,1 are Hilb (X3, 0) and its permutations. The center stratum is 3,1,1,1 Hilb (X3, 0) and the vertices correspond to strata obtained by 1,1,1,1 twisting Hilb (X3, 0). 46 r,s 3.3 The function ci,j depicted as an arrow from box (r, s) to box (i, j). 48 3.4 The distinguished arrows di,j and ui,j associated to box (i, j) in blue. 48 xiii 3.5 The arrow ui,j is not horizontal if and only if the box (i, j) is not top most in its column.
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