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Bounds on Hilbert Functions and Betti Numbers of Veronese Modules
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). (Algebraic Geometry)
2014 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The thesis is a collection of four papers dealing with Hilbert functions and Betti numbers.In the first paper, we study the h-vectors of reduced zero-dimensional schemes in  . In particular we deal with the problem of findingthe possible h-vectors for the union of two sets of points of given h-vectors. To answer to this problem, we give two kinds of bounds on theh-vectors and we provide an algorithm that calculates many possibleh-vectors.In the second paper, we prove a generalization of Green’s Hyper-plane Restriction Theorem to the case of finitely generated modulesover the polynomial ring, providing an upper bound for the Hilbertfunction of the general linear restriction of a module M in a degree dby the corresponding Hilbert function of a lexicographic module.In the third paper, we study the minimal free resolution of theVeronese modules, , where  by giving a formula for the Betti numbers in terms of the reduced homology of the squarefree divisor complex. We prove that is Cohen-Macaulay if and only if k < d, and that its minimal resolutionis linear when k > d(n − 1) − n. We prove combinatorially that the resolution of  is pure. We provide a formula for the Hilbert seriesof the Veronese modules. As an application, we calculate the completeBetti diagrams of the Veronese rings  .In the fourth paper, we apply the same combinatorial techniques inthe study of the properties of pinched Veronese rings, in particular weshow when this ring is Cohen-Macaulay. We also study the canonicalmodule of the Veronese modules.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2014. , vii, 31 p.
Series
TRITA-MAT-A, 2014:16
Keyword [en]
Hilbert function, Betti numbers, Veronese modules, Pinched veronese, h-vectors
National Category
Algebra and Logic Geometry
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-158913ISBN: 978-91-7595-394-6 (print)OAI: oai:DiVA.org:kth-158913DiVA: diva2:780001
Public defence
2015-02-04, F3, Lindstedtsvägen 26, KTH, Stockholm, 14:00 (English)
Opponent
Supervisors
Note

QC 20150115

Available from: 2015-01-15 Created: 2015-01-13 Last updated: 2015-01-15Bibliographically approved
List of papers
1. The h-vector of the union of two sets of points in the projective plane
Open this publication in new window or tab >>The h-vector of the union of two sets of points in the projective plane
2012 (English)In: Le Matematiche, ISSN 2037-5298, E-ISSN 0373-3505, Vol. 67, no 1Article in journal (Refereed) Published
Abstract [en]

Given two h-vectors, handh0, we study which are the possible h-vectors for the union of two disjoint sets of points in P2, respectively associated to h and h0and how they can be constructed. We will give some bounds for the resulting h-vector and we will show how to construct the minimal h-vector of the union among all possible ones.

Keyword
Union, Set, h-vector
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-133985 (URN)10.4418/2012.67.1.16 (DOI)
Note

QC 20131114

Available from: 2013-11-14 Created: 2013-11-14 Last updated: 2017-12-06Bibliographically approved
2. Green’s Hyperplane Restriction Theorem: an extension to modules
Open this publication in new window or tab >>Green’s Hyperplane Restriction Theorem: an extension to modules
2015 (English)In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 219, no 8, 3506-3517 p.Article in journal (Refereed) Published
Abstract [en]

In this paper, we prove a generalization of Green's Hyperplane RestrictionTheorem to the case of modules over the polynomial ring, providing in particularan upper bound for the Hilbert function of the general linear restrictionof a module M in a degree d by the corresponding Hilbert function of alexicographic module.

Keyword
Hilbert function, General linear restriction, Lexicographic modules
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-133986 (URN)10.1016/j.jpaa.2014.12.009 (DOI)000351979000025 ()2-s2.0-84925299702 (Scopus ID)
Note

Updated from manuscript to article.

QC 20150504

Available from: 2013-11-14 Created: 2013-11-14 Last updated: 2017-12-06Bibliographically approved
3. Syzygies of Veronese modules
Open this publication in new window or tab >>Syzygies of Veronese modules
(English)Manuscript (preprint) (Other academic)
Keyword
betti number, veronese, simplicial complex
National Category
Algebra and Logic Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-158911 (URN)
Note

QS 2015

Available from: 2015-01-13 Created: 2015-01-13 Last updated: 2015-01-15Bibliographically approved
4. Cohen-Macaulay Property of pinched Veronese Rings and Canonical Modules of Veronese  Modules
Open this publication in new window or tab >>Cohen-Macaulay Property of pinched Veronese Rings and Canonical Modules of Veronese  Modules
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper, we study the Betti numbers of pinched Veronese rings, by means of the reduced homology of the squarefree divisor complex. In particular, we study the Cohen-Macaulay property of these rings. Moreover, in the last section we compute the canonical modules of the Veronese modules.

Keyword
pinched veronese, cohen-macaulay, canonical module
National Category
Algebra and Logic
Identifiers
urn:nbn:se:kth:diva-158912 (URN)
Note

QS 2015

Available from: 2015-01-13 Created: 2015-01-13 Last updated: 2015-01-15Bibliographically approved

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