Mateev, Matey. The hvector of a standard determinantal scheme. 2014, PhD Thesis, University of Basel, Faculty of Science.

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Abstract
In this dissertation we study the hvector of a standard determinantal scheme
$X\subseteq\mathbb{P}^{n}$ via the corresponding degree matrix. We find simple
formulae for the length and the last entries of the hvector, as well as an
explicit formula for the hpolynomial. We also describe a recursive formula for
the hvector in terms of hvectors corresponding to submatrices of the degree
matrix of X. In codimension three we show that when the largest entry in the
degree matrix of X is sufficiently large and the first subdiagonal is entirely
positive the hvector of X is of decreasing type.
We prove that if a standard determinantal scheme is level, then its hvector is
a logconcave pure Osequence, and conjecture that the converse also holds.
Among other cases, we prove the conjecture in codimension two, or when the
entries of the corresponding degree matrix are positive.
We further investigate the combinatorial structure of the poset
$\mathcal{H}_{s}^{(t,c)}$ consisting of hvectors of length s, of codimension c
standard determinantal schemes, having degree matrices of size $t\times(t+c1)$
for some $t\geq1$. We show that
$\mathcal{H}_{s}^{(t,c)}$ obtains a natural
stratification, where each strata contains a maximum hvector. We prove
furthermore, that the only strata in which there exists also a minimum hvector
is the one consisting of hvectors of level standard determinantal schemes.
We also study posets of hvectors of standard determinantal ideals, which arise
from a matrix M, where the entries in each row have the same degree, and show
the existence of a minimum and a maximum hvector.
$X\subseteq\mathbb{P}^{n}$ via the corresponding degree matrix. We find simple
formulae for the length and the last entries of the hvector, as well as an
explicit formula for the hpolynomial. We also describe a recursive formula for
the hvector in terms of hvectors corresponding to submatrices of the degree
matrix of X. In codimension three we show that when the largest entry in the
degree matrix of X is sufficiently large and the first subdiagonal is entirely
positive the hvector of X is of decreasing type.
We prove that if a standard determinantal scheme is level, then its hvector is
a logconcave pure Osequence, and conjecture that the converse also holds.
Among other cases, we prove the conjecture in codimension two, or when the
entries of the corresponding degree matrix are positive.
We further investigate the combinatorial structure of the poset
$\mathcal{H}_{s}^{(t,c)}$ consisting of hvectors of length s, of codimension c
standard determinantal schemes, having degree matrices of size $t\times(t+c1)$
for some $t\geq1$. We show that
$\mathcal{H}_{s}^{(t,c)}$ obtains a natural
stratification, where each strata contains a maximum hvector. We prove
furthermore, that the only strata in which there exists also a minimum hvector
is the one consisting of hvectors of level standard determinantal schemes.
We also study posets of hvectors of standard determinantal ideals, which arise
from a matrix M, where the entries in each row have the same degree, and show
the existence of a minimum and a maximum hvector.
Advisors:  Gorla, Elisa 

Committee Members:  Brodmann, Markus 
Faculties and Departments:  05 Faculty of Science > Departement Mathematik und Informatik > Ehemalige Einheiten Mathematik & Informatik > Algebra (Gorla) 
UniBasel Contributors:  Mateev, Matey and Gorla, Elisa 
Item Type:  Thesis 
Thesis Subtype:  Doctoral Thesis 
Thesis no:  10803 
Thesis status:  Complete 
Bibsysno:  Link to catalogue 
Number of Pages:  84 S. 
Language:  English 
Identification Number: 

Last Modified:  22 Jan 2018 15:51 
Deposited On:  16 Jun 2014 07:07 
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