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Families of zero-cycles and divided powers: I. Representability
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).ORCID iD: 0000-0003-2505-6417
(English)Article in journal (Other academic) Submitted
Keyword [en]
Families of cycles, zero-cycles, divided powers, symmetric tensors, symmetric product, Chow scheme, Hilbert scheme
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-8706OAI: oai:DiVA.org:kth-8706DiVA: diva2:14099
Note
QC 20100908Available from: 2008-06-09 Created: 2008-06-09 Last updated: 2010-09-08Bibliographically approved
In thesis
1. Families of cycles and the Chow scheme
Open this publication in new window or tab >>Families of cycles and the Chow scheme
2008 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

The objects studied in this thesis are families of cycles on schemes. A space — the Chow variety — parameterizing effective equidimensional cycles was constructed by Chow and van der Waerden in the first half of the twentieth century. Even though cycles are simple objects, the Chow variety is a rather intractable object. In particular, a good functorial description of this space is missing. Consequently, descriptions of the corresponding families and the infinitesimal structure are incomplete. Moreover, the Chow variety is not intrinsic but has the unpleasant property that it depends on a given projective embedding. A main objective of this thesis is to construct a closely related space which has a good functorial description. This is partly accomplished in the last paper.

The first three papers are concerned with families of zero-cycles. In the first paper, a functor parameterizing zero-cycles is defined and it is shown that this functor is represented by a scheme — the scheme of divided powers. This scheme is closely related to the symmetric product. In fact, the scheme of divided powers and the symmetric product coincide in many situations.

In the second paper, several aspects of the scheme of divided powers are discussed. In particular, a universal family is constructed. A different description of the families as multi-morphisms is also given. Finally, the set of k-points of the scheme of divided powers is described. Somewhat surprisingly, cycles with certain rational coefficients are included in this description in positive characteristic.

The third paper explains the relation between the Hilbert scheme, the Chow scheme, the symmetric product and the scheme of divided powers. It is shown that the last three schemes coincide as topological spaces and that all four schemes are isomorphic outside the degeneracy locus.

The last paper gives a definition of families of cycles of arbitrary dimension and a corresponding Chow functor. In characteristic zero, this functor agrees with the functors of Barlet, Guerra, Kollár and Suslin-Voevodsky when these are defined. There is also a monomorphism from Angéniol's functor to the Chow functor which is an isomorphism in many instances. It is also confirmed that the morphism from the Hilbert functor to the Chow functor is an isomorphism over the locus parameterizing normal subschemes and a local immersion over the locus parameterizing reduced subschemes — at least in characteristic zero.

Place, publisher, year, edition, pages
Stockholm: KTH, 2008. v, 8 p.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 08-MA-06
Keyword
Families of cycles, Chow scheme, Hilbert scheme, zero-cycles, divided powers, symmetric tensors, symmetric product, Weil restriction, norm functor
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-4813 (URN)978-91-7178-999-0 (ISBN)
Public defence
2008-08-11, Nya kollegiesalen, F3, KTH, Lindstedtsvägen 26, Stockholm, 13:00
Opponent
Supervisors
Note
QC 20100908Available from: 2008-06-09 Created: 2008-06-09 Last updated: 2010-09-08Bibliographically approved

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