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Moduli of Surfaces and Applications to Curves

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Moduli of Surfaces and Applications to Curves Moduli of Surfaces and Applications to Curves Monica Marinescu Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2020 © 2020 Monica Marinescu All Rights Reserved Abstract Moduli of Surfaces and Applications to Curves Monica Marinescu This thesis has two parts. In the first part, we construct a moduli scheme F rns that parametrizes tuples pS1;S2;:::;Sn 1; p1; p2; : : : ; pnq where S1 is a fixed smooth surface over Spec R and Si 1 is the blowup of Si at a point pi, @1 ¤ i ¤ n. We show this moduli scheme is smooth and projective. We prove that F rns has smooth pnq @ ¤ ¤ ÞÑ divisors Di;j , 1 i j n, which correspond to tuples that map pj pi under the projection morphism Sj Ñ Si. When R k is an algebraically closed field, we ¦p r sq ¦p nq demonstrate that the Chow ring A F n is generated by these divisors over A S1 . ¦ We end by giving a precise description of A pF rnsq when S1 is a complex rational surface. In the second part of this thesis, we focus on finding a characterization of the smooth surfaces S on which a smooth very general curve of genus g embeds as an ample divisor. Our results can be summarized as follows: if the Kodaira dimension of S is κpSq ¡8 and S is not rational, then S is birational to C ¢ P1. If κpSq is 0 or 1, then such an embedding does not exist if the genus of C satisfies g ¥ 22. If κpSq 2 and the irregularity of S satisfies qpSq g, then S is birational to the symmetric square Sym2pCq. We analyze the conditions that need to be satisfied when S is a rational surface. The case in which S is of general type and qpSq 0 remains mainly open; however, we provide a partial answer to our question if S is a complete intersection. Contents Introduction 1 1 Definition of the moduli problem 12 2 Construction of the moduli scheme 16 3 Divisors of the moduli scheme 34 4 The Chow ring of the moduli scheme 41 5 The Chow ring of the moduli scheme for rational surfaces 47 6 A Question about Very General Curves 58 7 Curves and Surfaces 61 8 Preliminary results 66 9 Surfaces of Kodaira dimension ¡8 72 10 Surfaces of Kodaira dimension 0 86 10.1 Abelian Surfaces . 86 10.2 K3 Surfaces . 87 i 10.3 Enriques Surfaces . 89 10.4 Bielliptic Surfaces . 90 11 Surfaces of Kodaira dimension 1 91 12 Surfaces of Kodaira dimension 2 97 12.1 Surfaces of general type with qpSq g ................. 97 12.2 Surfaces of general type with qpSq 0 ................. 98 Bibliography 103 ii Acknowledgements I want to start by thanking my advisor Johan de Jong. I look up to Johan as a role model due to his incredible passion for mathematics, his vast knowledge, and his eagerness to share ideas with others. I learned a great amount from him through our many meetings, his research seminars, and the Stacks Project. I thank Johan for his incredible mentorship, through which I found confidence in my mathematical abilities. There were times when I struggled finding my way, and his guidance was key in helping me overcome the obstacles in my mathematical journey. I thank Robert Friedman, Henry Pinkham, Akash Sengupta and Daniel Litt for being part of my thesis committee, and for taking their time to carefully review this work and provide helpful feedback. I especially want to thank Daniel Litt for suggesting to me a project to think about back in 2018. His question and initial ideas were the starting point of this entire body of work. I am deeply grateful for everything I was offered in the Mathematics Department at Columbia, both academically and socially. These past five years have been a grand immersive experience into the world of mathematics research. Among the many avenues to learning, I am thankful for the multitude of seminars offered every semester, which helped me achieve a well-rounded perspective of mathematics. I thank the administrative staff, particularly Nathan Schweer, for all the hard work they put into making sure the graduate students are living their best life in the department. I am thankful for my academic brothers Raymond Cheng, Carl Lian, Dmitrii Pirozhkov, Noah Olander, Shizhang Li, Qixiao Ma, and Remy van Dobben de Bruyn. I learned a lot from our student seminars, our discussions, and from everyone's dif- ferent research topics. I particularly want to thank Raymond for reading some of my iii earlier work and giving me incredibly helpful comments. I want to thank many colleagues in the department who became lifelong friends, including: Clara Dolfen, Renata Picciotto, Raymond Cheng, Elena Giorgi, Laura Hayward, Stanislav Atanasov, Lea Kenigsberg, Sam Mundy, Noah Olander. You made these last five years some of the best of my life. You proved to me everyday that we can be happy in grad school. Lastly, I want to thank my family for being my support system: my parents, my sister, and my brother-in-law. I thank my partner, Owen, for making my life complete. iv Introduction Origin of the thesis The idea for this thesis started from Daniel Litt. He found Proposition 6.1 in Harris's and Mumford's paper \On the Kodaira Dimension of the Moduli Space of Curves" (see [20]), which states the following: if a surface S contains a very general curve of genus g ¥ 22 that moves in a non-trivial linear system, then S is birational to C ¢P1. Litt thought to analyze a similar question in a different setting: can we characterize the smooth surfaces on which a smooth very general curve C embeds as an ample divisor such that dim |C| 0? As a starting point, de Jong and Litt gave me the idea that Pic0pCq should be a simple abelian variety. To attack this question, we divided the problem into multiple cases, one for each minimal model of the surface S. This is the content of Chapters 6-12. Our result can be summarized as follows: if κpSq ¡8 and S is not rational, then S is birational to C ¢ P1. If the Kodaira dimension of S is 0 or 1, then such an embedding does not exists if the genus of C satisfies g ¥ 21. If κpSq 2 and qpSq g, then S is birational to the symmetric square Sym2pCq. There are a few cases left open. If S is a rational surface, we analyze the conditions that need to be satisfied in this situation (see Prop. 9.5 and 9.10). For the case in which S is of general type and qpSq 0, we prove the following partial result: if 1 S ãÑ Pr is a complete intersection and the composed morphism C ãÑ S ãÑ Pr satisfies the Maximal Rank Conjecture, then C is not ample on S if its genus is higher than 16. While working on the proofs above, we encountered the following scenario: say we have a series of morphisms Sn 1 Ñ ¤ ¤ ¤ Ñ S1, where S1 is a smooth minimal surface and each map Si 1 Ñ Si is the blowup of some point pi P Si. Say, for example, S1 is a K3 surface, so it has 19 moduli (see Thm. 10.2). Then Sn 1 has 19 2n moduli, since we add 2 moduli each time we blow up a new point. This line of thought made us consider the following scenario: say we fix a smooth surface S1 over Spec R. Consider all the ordered sequences of morphisms Sn 1 Ñ ¤ ¤ ¤ Ñ S1, where Si 1 Ñ Si is the blowup of Si at a point pi. Can we thoroughly construct a moduli space that parametrizes these objects? The answer is yes. In Chapters 1- 5, we construct a moduli scheme that parametrizes these sequences of blowups Sn 1 Ñ ¤ ¤ ¤ Ñ S1. We prove that the moduli functor is represented by a smooth projective scheme of dimension 2n over Spec R, as expected. We find smooth pnq p qn ÞÑ divisors Di;j that correspond to tuples Si; pi i1 where pj pi under the projection map Sj Ñ Si. When R k is an algebraically closed field, we prove that these ¦p r sq ¦p nq divisors generate the Chow ring A F n over A S1 . We end by giving a precise description of this Chow ring in the case where S1 is a complex rational surface. One of the resons why I found this problem interesting is that the moduli scheme has a very beautiful and natural construction, it behaves \as expected". All the intu- itive guesses we had about these moduli spaces and their properties while developing our theory turned out to be correct. Second of all, this construction is a new step forward in the study of moduli of surfaces. At the moment, one can find in the literature various constructions of moduli spaces parametrizing certain minimal surfaces. This moduli scheme stands 2 out because it parametrizes non-minimal surfaces. Lastly, looking at our results regarding the study of very general curves on smooth surfaces, here is what we can say based the open cases: if a curve C of high genus g is embedded on a smooth surface S as an ample divisor, then S is either a surface of general type of irregularity 0 (this is a vast class of surfaces), or it has a very specific blowup of P2.
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