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Families of cycles and the Chow scheme
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).ORCID iD: 0000-0003-2505-6417
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 [en]
Families of cycles, Chow scheme, Hilbert scheme, zero-cycles, divided powers, symmetric tensors, symmetric product, Weil restriction, norm functor
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-4813ISBN: 978-91-7178-999-0 (print)OAI: oai:DiVA.org:kth-4813DiVA: diva2:14103
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
List of papers
1. Families of zero-cycles and divided powers: I. Representability
Open this publication in new window or tab >>Families of zero-cycles and divided powers: I. Representability
(English)Article in journal (Other academic) Submitted
Keyword
Families of cycles, zero-cycles, divided powers, symmetric tensors, symmetric product, Chow scheme, Hilbert scheme
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-8706 (URN)
Note
QC 20100908Available from: 2008-06-09 Created: 2008-06-09 Last updated: 2010-09-08Bibliographically approved
2. Families of zero-cycles and divided powers: II. The universal family
Open this publication in new window or tab >>Families of zero-cycles and divided powers: II. The universal family
(English)Manuscript (Other academic)
Keyword
Families of cycles, zero-cycles, divided powers, universal family, Weil restriction, norm functor, Hilbert scheme
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-8707 (URN)
Note
QC 20100908Available from: 2008-06-09 Created: 2008-06-09 Last updated: 2010-09-08Bibliographically approved
3. Hilbert and Chow schemes of points, symmetric products and divided powers
Open this publication in new window or tab >>Hilbert and Chow schemes of points, symmetric products and divided powers
(English)Manuscript (Other academic)
Keyword
Divided powers, symmetric powers, Chow scheme, Hilbert scheme, Weighted projective spaces
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-8708 (URN)
Note
QC 20100908Available from: 2008-06-09 Created: 2008-06-09 Last updated: 2010-09-08Bibliographically approved
4. Families of cycles
Open this publication in new window or tab >>Families of cycles
(English)Manuscript (Other academic)
Keyword
Chow scheme, Hilbert scheme, divided powers, cycles, Weil divisors
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-8709 (URN)
Note
QC 20100908Available from: 2008-06-09 Created: 2008-06-09 Last updated: 2010-09-08Bibliographically approved

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