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The h-vector of the union of two sets of points in the projective plane
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
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.

Place, publisher, year, edition, pages
2012. Vol. 67, no 1
Keyword [en]
Union, Set, h-vector
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-133985DOI: 10.4418/2012.67.1.16OAI: oai:DiVA.org:kth-133985DiVA: diva2:664163
Note

QC 20131114

Available from: 2013-11-14 Created: 2013-11-14 Last updated: 2017-12-06Bibliographically approved
In thesis
1. Bounds on Hilbert Functions
Open this publication in new window or tab >>Bounds on Hilbert Functions
2013 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis is constituted of two articles, both related to Hilbert functions and h-vectors. In the first paper, we deal with h-vectorsof reduced zero-dimensional schemes in the projective plane, and, in particular, with the problem of finding the possible h-vectors for the union of two sets of points of given h-vectors. In the second paper, we generalize the Green’s Hyperplane Restriction Theorem to the case of modules over the polynomial ring.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2013. vii, 22 p.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 13:03
Keyword
Hilbert Functions, Commutative Algebra
National Category
Mathematics Algebra and Logic
Identifiers
urn:nbn:se:kth:diva-133942 (URN)978-91-7501-901-7 (ISBN)
Presentation
2013-11-11, 3418, Lindstedtsvägen 25, KTH, Stockholm, 10:00 (English)
Opponent
Supervisors
Note

QC 20131114

Available from: 2013-11-14 Created: 2013-11-13 Last updated: 2013-11-14Bibliographically approved
2. Bounds on Hilbert Functions and Betti Numbers of Veronese Modules
Open this publication in new window or tab >>Bounds on Hilbert Functions and Betti Numbers of Veronese Modules
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
Hilbert function, Betti numbers, Veronese modules, Pinched veronese, h-vectors
National Category
Algebra and Logic Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-158913 (URN)978-91-7595-394-6 (ISBN)
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

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