The h-vector of a standard determinantal scheme

In this dissertation we study the h-vector 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 h-vector, as well as an
explicit formula for the h-polynomial. We also describe a recursive formula for
the h-vector in terms of h-vectors 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 h-vector of X is of decreasing type.
We prove that if a standard determinantal scheme is level, then its h-vector is
a log-concave pure O-sequence, 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 h-vectors of length s, of codimension c
standard determinantal schemes, having degree matrices of size $t\times(t+c-1)$
for some $t\geq1$. We show that
$\mathcal{H}_{s}^{(t,c)}$ obtains a natural
stratification, where each strata contains a maximum h-vector. We prove
furthermore, that the only strata in which there exists also a minimum h-vector
is the one consisting of h-vectors of level standard determinantal schemes.
We also study posets of h-vectors 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 h-vector.