arXiv math/02090112002-09-02math.AG

行列式格式族的维数

Dimension of Families of Determinantal Schemes

Jan O. Kleppe, Rosa M. Miro-Roig

备注：: 37 pages

摘要：

余维$c$的一个方案$X\subset \PP^{n+c}$称为{ \em标准行列式}，如果它的齐次饱和理想可以由齐次$t \times (t+c-1)$矩阵的极大子式生成; $X$称为{ \em好行列式}，如果它是标准行列式和一般完全交.给定整数$a_0,a_1,...,a_{t+c-2}$和$b_1,...,b_t$，我们用$W(\underline{b};\underline{a})\subset \Hi ^p(\PP^{n+c})$表示（分别为$W_s(\underline{b};\underline{a})$）善的所在地（分别为由$t\times (t+c-1)$矩阵$(f_{ij})^{i=1,...,t}_{j=0,...,t+c-2}$的最大子式定义的余维$c$的标准）行列式方案$X\subset \PP^{n+c}$，其中$f_{ij}\in k[x_0,x_1,...,x_{n+c}]$是次数为$a_j-b_i$的齐次多项式。本文主要解决以下三个基本问题：（1）确定$W(\underline{b};\underline{a})$的维数（分别为：1）确定$W(\underline{b};\underline{a})$的维数; $W_s(\underline{b};\underline{a})$），（2）$W(\underline{b};\underline{a})$的闭包是否是$\Hi ^p(\PP^{n+c})$的不可约分支，（3）当$\Hi ^p(\PP^{n+c})$沿$W(\underline{b};\underline{a})$是一般光滑的时.关于问题（1），我们给出了$W(\underline{b};\underline{a})$维数的一个上界。$W_s(\underline{b};\underline{a})$），它适用于所有整数$a_0,a_1,...,a_{t+c-2}$和$b_1,...,b_t$，我们猜想这个界是尖锐的.对$2\le c\le 5$和$c\ge 6$在对$a_0,a_1,...,a_{t+c-2}$和$b_1,...,b_t$的某些限制下证明了该猜想.对于问题（2）和（3），我们对$2\le c \le 4$和$n\ge 2$以及在某些数值假设下对$c\ge 5$有肯定的回答。

英文摘要：

A scheme $X\subset \PP^{n+c}$ of codimension $c$ is called {\em standard determinantal} if its homogeneous saturated ideal can be generated by the maximal minors of a homogeneous $t \times (t+c-1)$ matrix and $X$ is said to be {\em good determinantal} if it is standard determinantal and a generic complete intersection. Given integers $a_0,a_1,...,a_{t+c-2}$ and $b_1,...,b_t$ we denote by $W(\underline{b};\underline{a})\subset \Hi ^p(\PP^{n+c})$ (resp. $W_s(\underline{b};\underline{a})$) the locus of good (resp. standard) determinantal schemes $X\subset \PP^{n+c}$ of codimension $c$ defined by the maximal minors of a $t\times (t+c-1)$ matrix $(f_{ij})^{i=1,...,t}_{j=0,...,t+c-2}$ where $f_{ij}\in k[x_0,x_1,...,x_{n+c}]$ is a homogeneous polynomial of degree $a_j-b_i$. In this paper we address the following three fundamental problems : To determine (1) the dimension of $W(\underline{b};\underline{a})$ (resp. $W_s(\underline{b};\underline{a})$) in terms of $a_j$ and $b_i$, (2) whether the closure of $W(\underline{b};\underline{a})$ is an irreducible component of $\Hi ^p(\PP^{n+c})$, and (3) when $\Hi ^p(\PP^{n+c})$ is generically smooth along $W(\underline{b};\underline{a})$. Concerning question (1) we give an upper bound for the dimension of $W(\underline{b};\underline{a})$ (resp. $W_s(\underline{b};\underline{a})$) which works for all integers $a_0,a_1,...,a_{t+c-2}$ and $b_1,...,b_t$, and we conjecture that this bound is sharp. The conjecture is proved for $2\le c\le 5$, and for $c\ge 6$ under some restriction on $a_0,a_1,...,a_{t+c-2}$ and $b_1,...,b_t$. For questions (2) and (3) we have an affirmative answer for $2\le c \le 4$ and $n\ge 2$, and for $c\ge 5$ under certain numerical assumptions.

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