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Moduli spaces of zero-dimensional geometric objects
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). (Algebra)
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The topic of this thesis is the study of moduli spaces of zero-dimensional geometricobjects. 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 parametrizesclosed subschemes of a fixed ambient scheme X. It has been known implicitly for sometime that the Hilbert scheme does not behave well when the scheme X is not separated.The article shows that the separation hypothesis is necessary in the sense thatthe component Hilb1(X) of Hilb(X) parametrizing subschemes of dimension zero andlength 1 does not exist if X is not separated.Article number two deals with the Chow scheme Chow 0,n(X) parametrizing zerodimensionaleffective cycles of length n on the given scheme X. There is a relatedconstruction, the Symmetric product Symn(X), defined as the quotient of the n-foldproduct X ×. . .×X of X by the natural action of the symmetric group Sn permutingthe factors. There is a canonical map Symn(X) " Chow0,n(X) that, set-theoretically,maps a tuple (x1, . . . , xn) to the cycle!nk=1 xk. In many cases this canonical map is anisomorphism. 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 symmetricproduct commutes with base change.The third article is related to the Fulton-MacPherson compactification of the configurationspace 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 schemeF(X, n) has a compactification X[n] which is obtained from the product Xn by a sequenceof blowups. Thus X[n] is itself not defined as a moduli space, but the pointson the boundary of X[n] may be interpreted as geometric objects called stable degenerations.It is then natural to ask if X[n] can be defined as a moduli space of stabledegenerations instead of as a blowup. In the third article we begin work towards ananswer to this question in the case where X = P2. We define a very general modulistack Xpv2 parametrizing projective schemes whose structure sheaf has vanishing secondcohomology. We then use Artin’s criteria to show that this stack is algebraic. Onemay define a stack SDX,n of stable degenerations of X and the goal is then to provealgebraicity of the stack SDX,n by using Xpv2.

Place, publisher, year, edition, pages
Stockholm: KTH , 2009. , vii, 10 p.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 09:09
Keyword [en]
Algebraic geometry, Commutative algebra, Moduli spaces
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-10662ISBN: 978-91-7415-379-8 (print)OAI: oai:DiVA.org:kth-10662DiVA: diva2:223079
Public defence
2009-08-17, Sydvästra Galleriet, KTH Biblioteket, Osquars backe 31, Stockholm, Stockholm, 13:00 (English)
Opponent
Supervisors
Note
QC 20100729Available from: 2009-06-12 Created: 2009-06-10 Last updated: 2010-07-29Bibliographically approved
List of papers
1. Non-effective deformations of Grothendieck’s Hilbertfunctor
Open this publication in new window or tab >>Non-effective deformations of Grothendieck’s Hilbertfunctor
2008 (English)In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 258, no 3, 513-519 p.Article in journal (Refereed) Published
Abstract [en]

Let X be a scheme that does not satisfy the valuativecriterion of separatedness. We show that the Hilbert functorparametrizing closed families of X that are flat, finite and of rankone is not represented by a scheme or an algebraic space.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-10664 (URN)10.1007/s00209-007-0183-7 (DOI)000252389200004 ()2-s2.0-38149095813 (Scopus ID)
Note
QC 20100729Available from: 2009-06-12 Created: 2009-06-12 Last updated: 2011-07-11Bibliographically approved
2. Counterexamples regarding symmetric tensors and divided powers
Open this publication in new window or tab >>Counterexamples regarding symmetric tensors and divided powers
2008 (English)In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 212, 2236-2249 p.Article in journal (Refereed) Published
Abstract [en]

We investigate the similarities and differences between the module of symmetric tensors TSAn (M) and the module of divided powers ΓAn (M). There is a canonical map ΓAn (M) → TSAn (M) which is an isomorphism in many important cases. We give examples showing that this map need neither be surjective nor injective in general. These examples also show that the functor TSAn does not in general commute with base change.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-10665 (URN)10.1016/j.jpaa.2008.03.024 (DOI)000257351900009 ()2-s2.0-44649141899 (Scopus ID)
Note
QC 20100729Available from: 2009-06-12 Created: 2009-06-12 Last updated: 2010-12-06Bibliographically approved
3. The stack of projective schemes with vanishing second cohomology
Open this publication in new window or tab >>The stack of projective schemes with vanishing second cohomology
(English)Manuscript (Other academic)
Identifiers
urn:nbn:se:kth:diva-10666 (URN)
Note
Preprint 2009Available from: 2009-06-12 Created: 2009-06-12 Last updated: 2010-04-28

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