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Hilbert Schemes and Rees Algebras

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Hilbert Schemes and Rees Algebras Hilbert schemes and Rees algebras GUSTAV SÆDÉN STÅHL Doctoral Thesis Stockholm, Sweden 2016 KTH TRITA-MAT-A 2016:11 Institutionen för Matematik ISRN KTH/MAT/A-16/11-SE 100 44 Stockholm ISBN 978-91-7729-171-8 SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen i matema- tik torsdagen den 8 december 2016 kl 13.00 i sal F3, Kungl Tekniska högskolan, Lindstedtsvägen 26, Stockholm. c Gustav Sædén Ståhl, 2016 • Tryck: Universitetsservice US AB iii Abstract The topic of this thesis is algebraic geometry, which is the mathematical subject that connects polynomial equations with geometric objects. Modern algebraic geometry has extended this framework by replacing polynomials with elements from a general commutative ring, and studies the geometry of abstract algebra. The thesis consists of six papers relating to some different topics of this field. The first three papers concern the Rees algebra. Given an ideal of a commutative ring, the corresponding Rees algebra is the coordinate ring of a blow-up in the subscheme defined by the ideal. We study a generalization of this concept where we replace the ideal with a module. In Paper A we give an intrinsic definition of the Rees algebra of a module in terms of divided powers. In Paper B we show that features of the Rees algebra can be explained by the theory of coherent functors. In Paper C we consider the geometry of the Rees algebra of a module, and characterize it by a universal property. The other three papers concern various moduli spaces. In Paper D we prove a partial generalization of Gotzmann’s persistence theorem to modules, and give explicit equations for the embedding of a Quot scheme inside a Grassmannian. In Paper E we expand on a result of Paper D, concerning the structure of certain Fitting ideals, to describe projective embeddings of open affine subschemes of a Hilbert scheme. Finally, in Paper F we introduce the good Hilbert functor parametrizing closed substacks with proper good moduli spaces of an algebraic stack, and we show that this functor is algebraic under certain conditions on the stack. iv Sammanfattning Ämnet för denna avhandling är algebraisk geometri, den matematiska teo- rin som kopplar samman polynomekvationer med geometriska objekt. Modern algebraisk geometri har vidareutvecklat denna idé genom att ersätta polynom med element från en godtycklig kommutativ ring, och betraktar abstrakt al- gebra från ett geometriskt perspektiv. Avhandlingen består av sex artiklar som studerar några olika ämnen i detta fält. De första tre artiklarna behandlar Reesalgebran. Givet ett ideal i en kom- mutativ ring så är motsvarande Reesalgebra koordinatringen av en uppblås- ning i delschemat definierat av idealet. Vi studerar en generalisering av denna konstruktion där vi ersatt idealet med en modul. I Artikel A ger vi en intrin- sisk definition av Reesalgebran av en modul i termer av dividerade potenser. I Artikel B visar vi att egenskaper hos Reesalgebran kan förklaras med hjälp av koherenta funktorer. I Artikel C studerar vi geometrin hos Reesalgebran av en modul, och karaktäriserar den med hjälp av en universell egenskap. De tre andra artiklarna behandlar olika typer av modulirum. I Artikel D visar vi en partiell generalisering av Gotzmanns persistenssats till moduler och ger explicita ekvationer för inbäddningen av ett Quotschema inuti en Grassmannian. I Artikel E vidareutvecklar vi ett resultat från Artikel D, rö- rande strukturen hos en viss typ av Fittingideal, för att ge beskrivningar av projektiva inbäddningar av öppna affina delscheman av ett Hilbertschema. Avslutningsvis inför vi i Artikel F den goda Hilbertfunktorn som parametri- serar slutna delstackar med propra goda modulirum av en algebraisk stack, och givet att stacken uppfyller vissa antaganden visar vi att denna funktor är algebraisk. Contents Contents v Acknowledgements vii Part I. Introduction and summary 1 From varieties to stacks 3 1.1 Varieties.................................. 3 1.2 Schemes.................................. 5 1.3 Algebraic spaces . 9 1.4 Algebraic stacks . 12 2 Introduction to the thesis 17 2.1 Group actions and gradings . 17 2.2 Bundlesandblow-ups .......................... 23 2.3 Projectiveresolutions........................... 27 2.4 Modulispaces............................... 29 2.5 Grothendieck’s existence theorem . 35 3 Summary of results 39 3.1 PaperA.................................. 39 3.2 PaperB.................................. 40 3.3 PaperC.................................. 41 3.4 PaperD.................................. 42 3.5 PaperE.................................. 43 3.6 PaperF.................................. 44 References 45 v vi CONTENTS Part II. Scientific papers Paper A An intrinsic definition of the Rees algebra of a module Proc. Edinb. Math. Soc. (2), to appear 17 pages Paper B Rees algebras of modules and coherent functors Preprint, arXiv:1409.6464 15 pages Paper C Total blow-ups of modules and universal flatifications Comm. Algebra, to appear 11 pages Paper D Gotzmann’s persistence theorem for finite modules J. Algebra, accepted (pending minor revisions) 14 pages Paper E Explicit projective embeddings of standard opens of the Hilbert scheme of points (joint with Roy Skjelnes) Preprint, arXiv:1605.07486 20 pages Paper F Good Hilbert functors Preprint, arXiv:1611.00976 16 pages Acknowledgements First and foremost, I want to express my deepest gratitude towards my advisor David Rydh for all the guidance, support, and patience. His mathematical insights and knowledge, and his willingness to help, are an inspiration. It has been an honor to learn mathematics from him. My second advisor and co-author Roy Skjelnes has given much help and encour- agement, for which I am very grateful. I have truly enjoyed discussing mathematics, and other things, with him. Moreover, I thank Mats Boij and the other members of the algebra and geometry group for making me feel included from day one. In particular, I would like to mention two people. Katharina Heinrich truly lived up to the role of an academic older sister by taking me under her wings when I was new and clueless, and her continued support and friendship have meant a lot. Anders Lundman has been my office mate, co-author, co-teacher and very good friend, and he made my time at the department much more enjoyable. Thank you both for all the good times. I have been a teacher of KTH:s Matematiska Cirkel for a majority of my time as a PhD student, which I have considered a great honor, and I want to thank my fellow teachers for the nice collaborations. I would also like to mention Dan Laksov who is sadly no longer with us. His positivity about me being a part of this course was very encouraging, and I hope that my work could live up to his legacy. These past years have also given me the opportunity to work with many great people. In particular, I thank Gaultier Lambert for organizing the graduate student seminar with me, my fellow organizers of GAeL XXI, and all my office mates. In addition, I thank Katharina Radermacher for the collaboration as PhD student representatives. More importantly I also want to thank her for all our conversations that have played a critical role in these past years, and have been fun, serious, meaningful, and necessary. During my time at the department I had the pleasure to meet many people that I now consider great friends, including everyone mentioned above. I also want to mention: Aron Wennman, Erik Aas, Gustav Behm, Joakim Roos, Olof Bergvall, Samuel Holmin, Sebastian Öberg, Simon Larson, and many more that I cannot fit on this page. You all made it fun to go to work. Finally, I thank my family for being there. vii Part I Introduction and summary 1 From varieties to stacks Algebraic geometry is the theory that relates algebraic objects (such as polynomials) to geometric objects (such as curves and surfaces). We will here give a brief and general introduction to some of the fundamental structures of this subject. These structures are used throughout all of algebraic geometry, and are not directly re- lated to the papers included in the second part of this thesis. A more specialized introduction to all the “abstract nonsense and formal GAGA” of the papers is instead presented in Chapter 2. We will start by considering polynomial equations and the classical notion of varieties. Then we go on to introduce schemes, which are generalizations of varieties and are the most central objects in modern algebraic geometry. After this, we go on to describe the even more general concepts of algebraic spaces and algebraic stacks. The first few pages should require little to no prerequisites, and is aimed at a general audience. As a rigorous text about all these concepts would be all too space consuming, we will however omit many details and refer the interested reader to the references given throughout the text. The amount of prerequisites required to follow along will therefore increase as the level of abstraction increases throughout the text, and the last sections are mostly aimed at graduate students interested in the topic. 1.1 Varieties The basic idea of algebraic geometry is to connect the solutions of polynomial equations with geometry. A solution to a polynomial equation, such as the equation x2 3xy +2y =0, ≠ can be realized as coordinates for a point in a space. For instance, x =2and y =1 is a solution to the equation above, since 22 3 2 1+2 1=0, and this solution ≠ · · · corresponds to a point with coordinates (2, 1) in a two-dimensional space. Affine varieties. An (affine) variety is the set of common solutions to a finite num- ber of polynomial equations. Below, we have drawn three varieties corresponding to three single polynomial equations over the real numbers. 3 4 CHAPTER 1. FROM VARIETIES TO STACKS 1 1 1 0 0 0 -1 -1 -1 -1 0 1 -1 0 1 -1 0 1 y2 = x2 + x3 x2 + y2 =1 (x2 + y2)3 =4x2y2 Note that when we depict the varieties as above we realize them as geometric objects.
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