Dimension of Families of Determinantal Schemes

Details Dimension of Families of Determinantal Schemes Dimension of Families of Determinantal Schemes

2002-09-02

arxiv.org/abs/math/0209011v1

Mathematics

Algebraic Geometry

37 pages

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Dimension of Families of Determinantal Schemes does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Dimension of Families of Determinantal Schemes, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Dimension of Families of Determinantal Schemes will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-280894