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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 $\mathbb{P}^{2}$ . 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, $S_{n,d,k}=\oplus_{i\geq 0} S_{k+id}$, where  $S = \mathbb{K}[x_1 , . . . , x_n ]$ by giving a formula for the Betti numbers in terms of the reduced homology of the squarefree divisor complex. We prove that $S_{n,d,k}$ 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 $S_{2,d,k}$ 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 $S_{3,4,0} , S_{3,5,0} and S_{4,3,0}$ .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
Mathematics
##### Identifiers
ISBN: 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)
##### 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
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
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
##### Keyword
betti number, veronese, simplicial complex
##### National Category
Algebra and Logic Geometry
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
##### 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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Cite
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