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California State University – Northridge

1. Chauhan, Sonia. Groebner basis and simplicial complexes attached to the tangent bundle of a family of determinantal varieties.

Degree: MS, Department of Mathematics, 2013, California State University – Northridge

URL: http://hdl.handle.net/10211.2/2509

For m C 2; consider the m??m determinantal variety of m??(m+1) matrices mod t2:
this is the variety Zm;m+1
m;2 obtained by considering generic m?? (m+ 1) matrices over
the ring F[t]~(t2), and setting the coe cients of powers of t of all m??m minors to
zero. The corresponding object mod t is of course the classical determinantal variety
Zm;m+1
m of m?? (m+ 1) matrices of rank less than m. We refer to the mod t2 variety
Zm;m+1
m;2 as the tangent bundle of the classical variety Zm;m+1
m.
In this thesis, we begin by providing a conjectured Groebner basis for the ideal
Im;m+1
m;2 which de nes Zm;m+1
m;2 , and as well, conjectured lead terms of the Groebner
basis. These conjectures were made based on explicit computations for the cases
2 B m B 6 that were done using the computer algebra system Singular ([8]). Since our
conjectured lead terms are squarefree for all m, we can construct Im, the Stanley-
Reisner simplicial complex attached to the ideal generated by our conjectured lead
terms. This complex has an existence of its own, independent of the conjectures, and
is an interesting object in its own right. The bulk of our thesis consists of the analysis
of this simplicial complex. For all values of m, we describe the facets of Im in terms
of their intersections with certain antidiagonals of a matrix of variables. Also for all
values of m, we derive a formula that counts the number of facets of this complex. This
number turns out to be the square of the degree of the classical variety Zm;m+1
m , which
yields evidence that our conjectures about the Groebner basis and their lead terms
are correct. In addition, for 2 B m B 6, we show that Im is shellable. Using standard
results, we conclude from the shellability of Im that for 2 B m B 6, the coordinate
ring of Zm;m+1
m;2 is Cohen-Macaulay, a property of great interest in algebraic geometry.
Further, for 2 B m B 6, we compute the Hilbert function of the Stanley-Reisner ring
of Im. The corresponding Hilbert series of Zm;m+1
m;2 pleasingly turns out to be the
square of the Hilbert series of the classical variety Zm;m+1
m , giving further credibility
to our conjectures.
*Advisors/Committee Members: Sethuraman, Bharath A. (advisor), Dye, John M. (committee member).*

Subjects/Keywords: Stanley Reisner complex; Dissertations, Academic – CSUN – Mathematics.

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Chauhan, S. (2013). Groebner basis and simplicial complexes attached to the tangent bundle of a family of determinantal varieties. (Masters Thesis). California State University – Northridge. Retrieved from http://hdl.handle.net/10211.2/2509

Chicago Manual of Style (16^{th} Edition):

Chauhan, Sonia. “Groebner basis and simplicial complexes attached to the tangent bundle of a family of determinantal varieties.” 2013. Masters Thesis, California State University – Northridge. Accessed June 16, 2019. http://hdl.handle.net/10211.2/2509.

MLA Handbook (7^{th} Edition):

Chauhan, Sonia. “Groebner basis and simplicial complexes attached to the tangent bundle of a family of determinantal varieties.” 2013. Web. 16 Jun 2019.

Vancouver:

Chauhan S. Groebner basis and simplicial complexes attached to the tangent bundle of a family of determinantal varieties. [Internet] [Masters thesis]. California State University – Northridge; 2013. [cited 2019 Jun 16]. Available from: http://hdl.handle.net/10211.2/2509.

Council of Science Editors:

Chauhan S. Groebner basis and simplicial complexes attached to the tangent bundle of a family of determinantal varieties. [Masters Thesis]. California State University – Northridge; 2013. Available from: http://hdl.handle.net/10211.2/2509

University of Kentucky

2. Stokes, Erik. THE h-VECTORS OF MATROIDS AND THE ARITHMETIC DEGREE OF SQUAREFREE STRONGLY STABLE IDEALS.

Degree: 2008, University of Kentucky

URL: http://uknowledge.uky.edu/gradschool_diss/636

Making use of algebraic and combinatorial techniques, we study two topics: the arithmetic degree of squarefree strongly stable ideals and the h-vectors of matroid complexes.
For a squarefree monomial ideal, I, the arithmetic degree of I is the number of facets of the simplicial complex which has I as its Stanley-Reisner ideal. We consider the case when I is squarefree strongly stable, in which case we give an exact formula for the arithmetic degree in terms of the minimal generators of I as well as a lower bound resembling that from the Multiplicity Conjecture. Using this, we can produce an upper bound on the number of minimal generators of any Cohen-Macaulay ideals with arbitrary codimension extending Dubreil’s theorem for codimension 2.
A matroid complex is a pure complex such that every restriction is again pure. It is a long-standing open problem to classify all possible h-vectors of such complexes. In the case when the complex has dimension 1 we completely resolve this question and we give some partial results for higher dimensions. We also prove the 1-dimensional case of a conjecture of Stanley that all matroid h-vectors are pure O-sequences. Finally, we completely characterize the Stanley-Reisner ideals of matroid complexes.

Subjects/Keywords: simplicial complex; matroid; h-vector; arithmetic degree; Stanley-Reisner ideal; Mathematics

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Stokes, E. (2008). THE h-VECTORS OF MATROIDS AND THE ARITHMETIC DEGREE OF SQUAREFREE STRONGLY STABLE IDEALS. (Doctoral Dissertation). University of Kentucky. Retrieved from http://uknowledge.uky.edu/gradschool_diss/636

Chicago Manual of Style (16^{th} Edition):

Stokes, Erik. “THE h-VECTORS OF MATROIDS AND THE ARITHMETIC DEGREE OF SQUAREFREE STRONGLY STABLE IDEALS.” 2008. Doctoral Dissertation, University of Kentucky. Accessed June 16, 2019. http://uknowledge.uky.edu/gradschool_diss/636.

MLA Handbook (7^{th} Edition):

Stokes, Erik. “THE h-VECTORS OF MATROIDS AND THE ARITHMETIC DEGREE OF SQUAREFREE STRONGLY STABLE IDEALS.” 2008. Web. 16 Jun 2019.

Vancouver:

Stokes E. THE h-VECTORS OF MATROIDS AND THE ARITHMETIC DEGREE OF SQUAREFREE STRONGLY STABLE IDEALS. [Internet] [Doctoral dissertation]. University of Kentucky; 2008. [cited 2019 Jun 16]. Available from: http://uknowledge.uky.edu/gradschool_diss/636.

Council of Science Editors:

Stokes E. THE h-VECTORS OF MATROIDS AND THE ARITHMETIC DEGREE OF SQUAREFREE STRONGLY STABLE IDEALS. [Doctoral Dissertation]. University of Kentucky; 2008. Available from: http://uknowledge.uky.edu/gradschool_diss/636

Dalhousie University

3. Connon, Emma. Generalizing Fröberg's Theorem on Ideals with Linear Resolutions.

Degree: PhD, Department of Mathematics & Statistics - Math Division, 2013, Dalhousie University

URL: http://hdl.handle.net/10222/38565

In 1990, Fröberg presented a combinatorial
classification of the quadratic square-free monomial ideals with
linear resolutions. He showed that the edge ideal of a graph has a
linear resolution if and only if the complement of the graph is
chordal. Since then, a generalization of Fröberg's theorem to
higher dimensions has been sought in order to classify all
square-free monomial ideals with linear resolutions. Such a
characterization would also give a description of all square-free
monomial ideals which are Cohen-Macaulay. In this thesis we explore
one method of extending Fröberg's result. We generalize the idea of
a chordal graph to simplicial complexes and use simplicial homology
as a bridge between this combinatorial notion and the algebraic
concept of a linear resolution. We are able to give a
generalization of one direction of Fröberg's theorem and, in
investigating the converse direction, find a necessary and
sufficient combinatorial condition for a square-free monomial ideal
to have a linear resolution over fields of characteristic
2.
*Advisors/Committee Members: Winfried Bruns (external-examiner), Sara Faridi (graduate-coordinator), Dorette Pronk (thesis-reader), Jason Brown (thesis-reader), Sara Faridi (thesis-supervisor), Not Applicable (ethics-approval), Yes (manuscripts), Not Applicable (copyright-release).*

Subjects/Keywords: monomial ideals; linear resolution; Fröberg's Theorem; chordal graph; Stanley-Reisner ideal; simplicial complex; simplicial homology

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Connon, E. (2013). Generalizing Fröberg's Theorem on Ideals with Linear Resolutions. (Doctoral Dissertation). Dalhousie University. Retrieved from http://hdl.handle.net/10222/38565

Chicago Manual of Style (16^{th} Edition):

Connon, Emma. “Generalizing Fröberg's Theorem on Ideals with Linear Resolutions.” 2013. Doctoral Dissertation, Dalhousie University. Accessed June 16, 2019. http://hdl.handle.net/10222/38565.

MLA Handbook (7^{th} Edition):

Connon, Emma. “Generalizing Fröberg's Theorem on Ideals with Linear Resolutions.” 2013. Web. 16 Jun 2019.

Vancouver:

Connon E. Generalizing Fröberg's Theorem on Ideals with Linear Resolutions. [Internet] [Doctoral dissertation]. Dalhousie University; 2013. [cited 2019 Jun 16]. Available from: http://hdl.handle.net/10222/38565.

Council of Science Editors:

Connon E. Generalizing Fröberg's Theorem on Ideals with Linear Resolutions. [Doctoral Dissertation]. Dalhousie University; 2013. Available from: http://hdl.handle.net/10222/38565