In two dimensional regular local rings integrally closed ideals have a unique factorization property and have a Cohen-Macaulay associated graded ring. In higher dimension these properties do not hold for general integrally closed ideals and the goal
of the paper is to identify a subclass of integrally closed ideals for which they do. We restrict our attention to 0-dimensional homogeneous ideals in polynomial rings $R$ of arbitrary dimension and identify a class of integrally closed ideals, the Goto-class $G^*$, that is closed under product and that has a suitable unique factorization property. Ideals in $G^*$ have a Cohen-Macaulay associated graded ring if either they are monomial or $dim Rleq 3$. Our approach is based on the study of the relationship between the notions of integrally closed, contracted, full and componentwise linear ideals.
Let $R=oplus_{igeq 0} R_i$ be an Artinian standard graded $K$-algebra defined by quadrics. Assume that $dim R_2leq 3$ and that $K$ is algebraically closed of characteristic $ eq 2$. We show that $R$ is defined by a Grobner basis of quadrics with, ess
entially, one exception. The exception is given by $K[x,y,z]/I$ where $I$ is a complete intersection of 3 quadrics not containing the square of a linear form.
Let $K$ be a field and $V$ and $W$ be $K$-vector spaces of dimension $m$ and $n$. Let $phi$ be the canonical map from $Hom(V,W)$ to $Hom(wedge^t V,wedge^t W)$. We investigate the Zariski closure $X_t$ of the image $Y_t$ of $phi$. In the case $t=min(m
,n)$, $Y_t=X_t$ is the cone over a Grassmannian, but $X_t$ is larger than $Y_t$ for $1<t<min(m,n)$. We analyze the $G=GL(V)timesGL(W)$-orbits in $X_t$ via the corresponding $G$-stable prime ideals. It turns out that they are classified by two numerical invariants, one of which is the rank and the other a related invariant that we call small rank. Surprisingly, the orbits in $X_tsetminus Y_t$ arise from the images $Y_u$ for $u<t$ and simple algebraic operations. In the last section we determine the singular locus of $X_t$. Apart from well-understood exceptional cases, it is formed by the elements of rank $le 1$ in $Y_t$.
The paper has two goals: the study the associated graded ring of contracted homogeneous ideals in $K[x,y]$ and the study of the Groebner fan of the ideal $P$ of the rational normal curve in ${bf P}^d$. These two problems are, quite surprisingly, very
tightly related. We completely classify the contracted ideals with a Cohen-Macaulay associated graded rings in terms of the numerical invariants arising from Zariskis factorization. We determine explicitly all the initial ideals (monomial or not) of $P$ that are Cohen-Macaulay.
An Artinian ideal $I$ of $k[x,y]$ has many Hilbert-Burch matrices. We show that there is a canonical choice. As an application, we determine the dimension of certain affine Grobner cells and their Betti strata recovering results of Ellingsrud and Str{o}mme, Gottsche and Iarrobino.
