ﻻ يوجد ملخص باللغة العربية
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 $Y_t$.
Given an ideal $I=(f_1,ldots,f_r)$ in $mathbb C[x_1,ldots,x_n]$ generated by forms of degree $d$, and an integer $k>1$, how large can the ideal $I^k$ be, i.e., how small can the Hilbert function of $mathbb C[x_1,ldots,x_n]/I^k$ be? If $rle n$ the sma
We call shifted power a polynomial of the form $(x-a)^e$. The main goal of this paper is to obtain broadly applicable criteria ensuring that the elements of a finite family $F$ of shifted powers are linearly independent or, failing that, to give a lo
In this article, we study the regularity of integral closure of powers of edge ideals. We obtain a lower bound for the regularity of integral closure of powers of edge ideals in terms of induced matching number of graphs. We prove that the regularity
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