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Bounds on Hilbert Functions and Betti Numbers of Veronese Modules
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.). (Algebraic Geometry)
2014 (engelsk)Doktoravhandling, med artikler (Annet vitenskapelig)
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.

sted, utgiver, år, opplag, sider
Stockholm: KTH Royal Institute of Technology, 2014. , s. vii, 31
Serie
TRITA-MAT-A ; 2014:16
Emneord [en]
Hilbert function, Betti numbers, Veronese modules, Pinched veronese, h-vectors
HSV kategori
Forskningsprogram
Matematik
Identifikatorer
URN: urn:nbn:se:kth:diva-158913ISBN: 978-91-7595-394-6 (tryckt)OAI: oai:DiVA.org:kth-158913DiVA, id: diva2:780001
Disputas
2015-02-04, F3, Lindstedtsvägen 26, KTH, Stockholm, 14:00 (engelsk)
Opponent
Veileder
Merknad

QC 20150115

Tilgjengelig fra: 2015-01-15 Laget: 2015-01-13 Sist oppdatert: 2015-01-15bibliografisk kontrollert
Delarbeid
1. The h-vector of the union of two sets of points in the projective plane
Åpne denne publikasjonen i ny fane eller vindu >>The h-vector of the union of two sets of points in the projective plane
2012 (engelsk)Inngår i: Le Matematiche, ISSN 2037-5298, E-ISSN 0373-3505, Vol. 67, nr 1Artikkel i tidsskrift (Fagfellevurdert) 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.

Emneord
Union, Set, h-vector
HSV kategori
Identifikatorer
urn:nbn:se:kth:diva-133985 (URN)10.4418/2012.67.1.16 (DOI)
Merknad

QC 20131114

Tilgjengelig fra: 2013-11-14 Laget: 2013-11-14 Sist oppdatert: 2017-12-06bibliografisk kontrollert
2. Green’s Hyperplane Restriction Theorem: an extension to modules
Åpne denne publikasjonen i ny fane eller vindu >>Green’s Hyperplane Restriction Theorem: an extension to modules
2015 (engelsk)Inngår i: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 219, nr 8, s. 3506-3517Artikkel i tidsskrift (Fagfellevurdert) 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.

Emneord
Hilbert function, General linear restriction, Lexicographic modules
HSV kategori
Identifikatorer
urn:nbn:se:kth:diva-133986 (URN)10.1016/j.jpaa.2014.12.009 (DOI)000351979000025 ()2-s2.0-84925299702 (Scopus ID)
Merknad

Updated from manuscript to article.

QC 20150504

Tilgjengelig fra: 2013-11-14 Laget: 2013-11-14 Sist oppdatert: 2017-12-06bibliografisk kontrollert
3. Syzygies of Veronese modules
Åpne denne publikasjonen i ny fane eller vindu >>Syzygies of Veronese modules
(engelsk)Manuskript (preprint) (Annet vitenskapelig)
Emneord
betti number, veronese, simplicial complex
HSV kategori
Forskningsprogram
Matematik
Identifikatorer
urn:nbn:se:kth:diva-158911 (URN)
Merknad

QS 2015

Tilgjengelig fra: 2015-01-13 Laget: 2015-01-13 Sist oppdatert: 2015-01-15bibliografisk kontrollert
4. Cohen-Macaulay Property of pinched Veronese Rings and Canonical Modules of Veronese  Modules
Åpne denne publikasjonen i ny fane eller vindu >>Cohen-Macaulay Property of pinched Veronese Rings and Canonical Modules of Veronese  Modules
(engelsk)Manuskript (preprint) (Annet vitenskapelig)
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.

Emneord
pinched veronese, cohen-macaulay, canonical module
HSV kategori
Identifikatorer
urn:nbn:se:kth:diva-158912 (URN)
Merknad

QS 2015

Tilgjengelig fra: 2015-01-13 Laget: 2015-01-13 Sist oppdatert: 2015-01-15bibliografisk kontrollert

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Totalt: 375 treff
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