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Moduli Spaces of Zero-Dimensional Geometric Objects

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Moduli Spaces of Zero-Dimensional Geometric Objects Moduli spaces of zero-dimensional geometric objects CHRISTIAN LUNDKVIST Doctoral Thesis Stockholm, Sweden 2009 TRITA-MAT-09-MA-09 ISSN 1401-2278 KTH Matematik ISRN KTH/MAT/DA 09/06-SE SE-100 44 Stockholm ISBN 978-91-7415-379-8 SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till of- fentlig granskning för avläggande av teknologie doktorsexamen i matematik måndagen den 17 augusti 2009 klockan 13.00 i Sydvästra galleriet, KTH biblioteket, Kungl Tekniska hög- skolan, Osquars backe 31, Stockholm. © Christian Lundkvist, maj 2009 Tryck: Universitetsservice US AB iii Abstract The topic of this thesis is the study of moduli spaces of zero-dimensional geometric objects. The thesis consists of three articles each focusing on a particular moduli space. The first article concerns the Hilbert scheme Hilb(X). This moduli space parametrizes closed subschemes of a fixed ambient scheme X. It has been known implicitly for some time that the Hilbert scheme does not behave well when the scheme X is not sepa- rated. The article shows that the separation hypothesis is necessary in the sense that the component Hilb1(X) of Hilb(X) parametrizing subschemes of dimension zero and length 1 does not exist if X is not separated. Article number two deals with the Chow scheme Chow0,n(X) parametrizing zero- dimensional effective cycles of length n on the given scheme X. There is a related construction, the Symmetric product Symn(X), defined as the quotient of the n-fold product X ... X of X by the natural action of the symmetric group Sn permuting × × n the factors. There is a canonical map Sym (X) Chow0,n(X) that, set-theoretically, n → maps a tuple (x1, . , xn) to the cycle k=1 xk. In many cases this canonical map is an isomorphism. We explore in this paper some examples where it is not an isomorphism. This will also lead to some results concerning! the question whether the symmetric product commutes with base change. The third article is related to the Fulton-MacPherson compactification of the config- uration space of points. Here we begin by considering the configuration space F (X, n) parametrizing n-tuples of distinct ordered points on a smooth scheme X. The scheme F (X, n) has a compactification X[n] which is obtained from the product Xn by a se- quence of blowups. Thus X[n] is itself not defined as a moduli space, but the points on the boundary of X[n] may be interpreted as geometric objects called stable degen- erations. It is then natural to ask if X[n] can be defined as a moduli space of stable degenerations instead of as a blowup. In the third article we begin work towards an answer to this question in the case where X = P2. We define a very general moduli stack Xpv2 parametrizing projective schemes whose structure sheaf has vanishing sec- ond cohomology. We then use Artin’s criteria to show that this stack is algebraic. One may define a stack X,n of stable degenerations of X and the goal is then to prove SD algebraicity of the stack X,n by using X . SD pv2 iv Sammanfattning Denna avhandling behandlar modulirum av nolldimensionella geometriska objekt. Avhandlingen består av tre artiklar som var och en fokuserar på ett speciellt moduli- rum. Den första artikeln tar upp Hilbertschemat Hilb(X). Detta modulirum parametrise- rar slutna delscheman av ett fixt schema X. Vanligtvis studeras Hilbertschemat utifrån hypotesen att schemat X är separerat. Artikeln visar att denna hypotes är nödvän- dig eftersom komponenten Hilb1(X) av Hilb(X) som parametriserar delscheman av dimension noll och längd ett ej existerar om X inte är separerat. Den andra artikeln handlar om Chowschemat Chow0,n(X) som parametriserar nolldimensionella cykler av längd n på ett givet schema X. En relaterad konstruk- tion är den symmetriska produkten Symn(X) som defineras som kvoten av produkten X ... X under verkan av den symmetriska gruppen Sn som permuterar faktorer- × × n na. Det finns en kanonisk avbildning Sym (X) Chow0,n(X) som mängdteoretiskt n → avbildar en tupel (x1, . , xn) på cykeln k=1 xk. Denna kanoniska avbildning är i många fall en isomorfi. I artikeln i fråga utforskar vi ett antal exempel där den kano- niska avbildningen inte är en isomorfi. Detta! leder även till några resultat angående frågan om den symmetriska produkten kommuterar med basbyte. Den tredje artikeln har kopplingar till Fulton-MacPhersons kompaktifiering av kon- figurationsrummet av punkter. Här börjar vi med att studera konfigurationsrummet F (X, n) som parametriserar n-tupler av distinkta ordnade punkter på ett givet ic- kesingulärt schema X. Schemat F (X, n) har en kompaktifiering X[n] som fås från produkten Xn genom en följd av uppblåsningar. Därmed är X[n] i sig inte definierat som ett modulirum, men punkterna på randen till X[n] kan tolkas som geometriska objekt, så kallade stabila degenerationer. Det är därför naturligt att fråga sig om X[n] kan definieras som ett modulirum av stabila degenerationer istället för att konstrueras genom uppblåsningar. I artikel nummer tre påbörjar vi arbetet mot svaret till denna 2 fråga i fallet X = P . Vi definierar en generell modulistack Xpv2 som parametriserar projektiva scheman vars strukturkärve har kohomologi noll i grad 2. Vi använder där- efter Artins kriterier för att visa att stacken Xpv2 är algebraisk. Vi definierar en stack X,n som parametriserar stabila degenerationer och förhoppningen är att kunna an- SD vända X för att visa att X,n är algebraisk. pv2 SD Contents Contents v Acknowledgements vii I Introduction and Summary 1 1 Introduction 3 1.1 Moduli spaces . 3 2 Summary 5 2.1 Paper A - The Hilbert scheme . 5 2.2 Paper B - The Chow scheme . 5 2.3 Paper C - The Fulton-MacPherson compactification . 6 Bibliography 9 II Scientific Papers 11 A Non-effective deformations of Grothendieck’s Hilbert functor (with R. Skjelnes) Mathematische Zeitschrift, Vol. 258, Nr. 3, March 2008. B Counterexamples regarding symmetric tensors and divided powers Journal of Pure and Applied Algebra, Vol. 212, Issue 10, October 2008. C The stack of projective schemes with vanishing second cohomology Preprint, May 2009. v Acknowledgements I would first like to thank my advisor Dan Laksov for his support and enthusiasm in this endeavor. One of Dan’s main strengths is that he is always very honest and speaks his mind. Thus when he gives criticism it is always justified and when he gives praise it is never empty words. A big thanks to my co-author Roy Skjelnes and to the Algebra group at KTH for a warm, friendly atmosphere and many stimulating fika-discussions. I am also grateful to the University of Michigan math department where I spent the academic year 2007 – 2008. I particularly wish to thank Mattias Jonsson who invited me and with whom I had the pleasure of working. I am also indebted to Professor William Fulton for several interesting conversations about the topics covered in this thesis. For many relaxing (and also wild!) evenings at the various pubs and bars in Ann Arbor I would like to thank the math and physics graduate students at UMich, especially Rodrigo Parra. I have been fortunate enough to have made many friends during my years at KTH. A big thanks to all my fellow graduate students at the KTH math department who have made my graduate years a real pleasure. Extra thanks to my officemates Erik Lindgren, Christopher Svedberg and in particular Joanna Nilsson whom I consider a very close friend. Also thanks to my mathematical sibling David Rydh who always found the time to share his mathematical insights. Lots of love goes to the Täby crew: Anders & Azza, Better, Bebbe, Noonan, Andreas & Jenny, Nina & Magnus, Vera & Neso, Jonas & Ann and everyone else in this big family. As long as you are my friends I will always feel like I have the Quad. Maria — keep living the dream! Tomas, you and I have been friends now for over 20 years, and I hope we will remain friends for at least another 200. My parents have provided a loving home and for this I will always be grateful. In particular they fought hard to provide me with the best possible education, and they have always supported me when I needed it. My brother Peter and I have shared a good childhood with the occasional sibling rivalry. Thanks for putting up with me when I was an annoying lillebror. Finally, to Stephanie I would just like to say thank you for your patience during the writing of this thesis. As for the other things I want to tell you, I hope my everyday actions will convey my feelings better than words written here. vii Part I Introduction and Summary 1 Chapter 1 Introduction 1.1 Moduli spaces This thesis is concerned with the field within algebraic geometry called moduli spaces. The basic idea is the following: We are interested in the behaviour of certain geometric objects; these can be for instance closed subvarieties of a fixed variety, collections of n distinct ordered points on a fixed variety, or more general objects such as smooth curves of a fixed genus. We then try to find a space X which parametrizes these objects; this means that a point of X corresponds to one of these objects, a curve on X corresponds to a smoothly varying one-dimensional family of objects, etc. The correspondence is supposed to be “natural” or “functorial” in the following sense: We may consider a functor F from varieties to sets by choosing F (V ) to be the set of families of objects indexed by the variety V .
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