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Hilbert schemes and Rees algebras
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0001-8893-5211
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The topic of this thesis is algebraic geometry, which is the mathematical subject that connects polynomial equations with geometric objects. Modern algebraic geometry has extended this framework by replacing polynomials with elements from a general commutative ring, and studies the geometry of abstract algebra. The thesis consists of six papers relating to some different topics of this field.

The first three papers concern the Rees algebra. Given an ideal of a commutative ring, the corresponding Rees algebra is the coordinate ring of a blow-up in the subscheme defined by the ideal. We study a generalization of this concept where we replace the ideal with a module. In Paper A we give an intrinsic definition of the Rees algebra of a module in terms of divided powers. In Paper B we show that features of the Rees algebra can be explained by the theory of coherent functors. In Paper C we consider the geometry of the Rees algebra of a module, and characterize it by a universal property.

The other three papers concern various moduli spaces. In Paper D we prove a partial generalization of Gotzmann’s persistence theorem to modules, and give explicit equations for the embedding of a Quot scheme inside a Grassmannian. In Paper E we expand on a result of Paper D, concerning the structure of certain Fitting ideals, to describe projective embeddings of open affine subschemes of a Hilbert scheme. Finally, in Paper F we introduce the good Hilbert functor parametrizing closed substacks with proper good moduli spaces of an algebraic stack, and we show that this functor is algebraic under certain conditions on the stack. 

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2016. , vii, 49 p.
Series
TRITA-MAT-A, 2016:11
National Category
Algebra and Logic Geometry
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-195717ISBN: 978-91-7729-171-8OAI: oai:DiVA.org:kth-195717DiVA: diva2:1045297
Public defence
2016-12-08, F3, Lindstedtsvägen 26, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

QC 20161110

Available from: 2016-11-10 Created: 2016-11-08 Last updated: 2016-11-10Bibliographically approved
List of papers
1. An intrinsic definition of the Rees algebra of a module
Open this publication in new window or tab >>An intrinsic definition of the Rees algebra of a module
(English)In: Proceedings of the Edinburgh Mathematical Society, ISSN 0013-0915, E-ISSN 1464-3839Article in journal (Refereed) Accepted
Abstract [en]

This paper concerns a generalization of the Rees algebra of ideals due to Eisenbud, Huneke and Ulrich that works for any finitely generated module over a noetherian ring. Their definition is in terms of maps to free modules. We give an intrinsic definition using divided powers.

National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-156533 (URN)
Note

QCR 20161110

Available from: 2014-12-01 Created: 2014-11-29 Last updated: 2016-11-10Bibliographically approved
2. Rees algebras of modules and coherent functors
Open this publication in new window or tab >>Rees algebras of modules and coherent functors
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We show that several properties of the theory of Rees algebras of modules become more transparent using the category of coherent functors rather than working directly with modules. In particular, we show that the Rees algebra is induced by a canonical map of coherent functors.

National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-156534 (URN)
Note

QCR 20161110

Available from: 2014-12-01 Created: 2014-11-29 Last updated: 2016-11-10Bibliographically approved
3. Total blow-ups of modules and universal flatifications
Open this publication in new window or tab >>Total blow-ups of modules and universal flatifications
(English)In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125Article in journal (Refereed) Accepted
Abstract [en]

We study the projective spectrum of the Rees algebra of a module, and characterize it by a universal property. As applications, we give descriptions of universal flatifications of modules and of birational projective morphisms.

National Category
Algebra and Logic Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-195714 (URN)10.1080/00927872.2016.1244270 (DOI)
Note

QCR 20161110

Available from: 2016-11-08 Created: 2016-11-08 Last updated: 2016-11-10Bibliographically approved
4. Gotzmann's persistence theorem for finite modules
Open this publication in new window or tab >>Gotzmann's persistence theorem for finite modules
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We prove a generalization of Gotzmann's persistence theorem in the case of modules with constant Hilbert polynomial. As a consequence, we show that the defining equations that give the embedding of a Quot scheme of points into a Grassmannian are given by a single Fitting ideal.

National Category
Algebra and Logic Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-156535 (URN)
Note

QCR 20161110

Available from: 2014-12-01 Created: 2014-11-29 Last updated: 2016-11-10Bibliographically approved
5. Explicit projective embeddings of standard opens of the Hilbert scheme of points
Open this publication in new window or tab >>Explicit projective embeddings of standard opens of the Hilbert scheme of points
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We describe explicitly how certain standard opens of the Hilbert scheme of points are embedded into Grassmannians. The standard opens of the Hilbert scheme that we consider are given as the intersection of a corresponding basic open affine of the Grassmannian and a closed stratum determined by a Fitting ideal.

National Category
Algebra and Logic Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-195715 (URN)
Note

QCR 20161110

Available from: 2016-11-08 Created: 2016-11-08 Last updated: 2016-11-10Bibliographically approved
6. Good Hilbert functors
Open this publication in new window or tab >>Good Hilbert functors
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We introduce the good Hilbert functor and prove that it is algebraic. This functor generalizes various versions of the Hilbert moduli problem, such as the multigraded Hilbert scheme and the invariant Hilbert scheme. Moreover, we generalize a result concerning formal GAGA for good moduli spaces.

National Category
Algebra and Logic Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-195716 (URN)
Note

QCR 20161110

Available from: 2016-11-08 Created: 2016-11-08 Last updated: 2016-11-10Bibliographically approved

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