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Counterexamples regarding symmetric tensors and divided powers
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
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.

Place, publisher, year, edition, pages
2008. Vol. 212, 2236-2249 p.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-10665DOI: 10.1016/j.jpaa.2008.03.024ISI: 000257351900009Scopus ID: 2-s2.0-44649141899OAI: oai:DiVA.org:kth-10665DiVA: diva2:223372
Note
QC 20100729Available from: 2009-06-12 Created: 2009-06-12 Last updated: 2017-12-13Bibliographically approved
In thesis
1. Moduli spaces of zero-dimensional geometric objects
Open this publication in new window or tab >>Moduli spaces of zero-dimensional geometric objects
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
Algebraic geometry, Commutative algebra, Moduli spaces
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-10662 (URN)978-91-7415-379-8 (ISBN)
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

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