On the analytic spread and the reduction number of the ideal of maximal minors

Abstract

Let $m$, $n$, $a_1$, ..., $a_r$, $b_1$, ..., $b_r$ be integers with $1\le a_1<...<a_r\le m$ and $1\le b_1<...<b_r\le n$. And let $x$ be the universal $m\times n$ matrix with the property that $i$-minors of first $a_i-1$ rows and first $b_i-1$ columns are all zero, for $i=1$, ..., $r+1$ ($a_{r+1}=m+1$ and $b_{r+1}=n+1$). For an integer $u$ with $1\le u\le m$, we denote by $U$ the $u\times n$ matrix consisting of the first $u$ rows of $x$. In this paper, we consider the analytic spread and the reduction number of the ideal of maximal minors of $U$