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Title: | Dimension of families of determinantal schemes |
Author: | Kleppe, J.O. Miró-Roig, Rosa M. (Rosa Maria) |
Keywords: | Geometria algebraica Esquemes (Geometria algebraica) Algebraic geometry Schemes (Algebraic geometry) |
Issue Date: | 2005 |
Publisher: | American Mathematical Society (AMS) |
Abstract: | A scheme $X\subset \mathbb{P} ^{n+c}$ of codimension $c$ is called 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 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 $W(\underline{b};\underline{a})\subset \operatorname{Hilb} ^p(\mathbb{P} ^{n+c})$(resp. $W_s(\underline{b};\underline{a})$) the locus of good (resp. standard) determinantal schemes $X\subset \mathbb{P} ^{n+c}$ 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 $\operatorname{Hilb} ^p(\mathbb{P} ^{n+c})$, and (3) when $\operatorname{Hilb} ^p(\mathbb{P} ^{n+c})$ 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. |
Note: | Reproducció del document publicat a: |
It is part of: | Transactions of the American Mathematical Society, 2005, vol. 357, num. 7 |
URI: | http://hdl.handle.net/2445/96560 |
ISSN: | 0002-9947 |
Appears in Collections: | Articles publicats en revistes (Matemàtiques i Informàtica) |
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