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Dimension of families of determinantal schemes

Authors:	Jan O. Kleppe Rosa M. Miró -Roig

Affiliation:	Faculty of Engineering, Oslo University College, Cort Adelers gt. 30, N-0254 Oslo, Norway ; Facultat de Matemàtiques, Departament d'Algebra i Geometria, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain

Abstract:	A scheme $Xsubset mathbb{P} ^{n+c}$ of codimension is called standard determinantal if its homogeneous saturated ideal can be generated by the maximal minors of a homogeneous matrix and 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 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 $Xsubset mathbb{P} ^{n+c}$ of codimension defined by the maximal minors of a 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 . In this paper we address the following three fundamental problems: To determine (1) the dimension of (resp. $W_s(underline{b};underline{a})$ ) in terms of and , (2) whether the closure of 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 . Concerning question (1) we give an upper bound for the dimension of (resp. $W_s(underline{b};underline{a})$ ) which works for all integers $a_0,a_1,...,a_{t+c-2}$ and , and we conjecture that this bound is sharp. The conjecture is proved for , and for under some restriction on $a_0,a_1,...,a_{t+c-2}$ and . For questions (2) and (3) we have an affirmative answer for and , and for under certain numerical assumptions.

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