Dimension of families of determinantal schemes
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- by Jan O. Kleppe and Rosa M. Miró-Roig PDF
- Trans. Amer. Math. Soc. 357 (2005), 2871-2907 Request permission
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.References
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Additional Information
- Jan O. Kleppe
- Affiliation: Faculty of Engineering, Oslo University College, Cort Adelers gt. 30, N-0254 Oslo, Norway
- Email: JanOddvar.Kleppe@iu.hio.no
- Rosa M. Miró-Roig
- Affiliation: Facultat de Matemàtiques, Departament d’Algebra i Geometria, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain
- MR Author ID: 125375
- ORCID: 0000-0003-1375-6547
- Email: miro@ub.edu
- Received by editor(s): August 1, 2003
- Received by editor(s) in revised form: December 23, 2003
- Published electronically: December 9, 2004
- Additional Notes: The second author was partially supported by BFM2001-3584
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 357 (2005), 2871-2907
- MSC (2000): Primary 14M12, 14C05, 14H10, 14J10; Secondary 14N05
- DOI: https://doi.org/10.1090/S0002-9947-04-03648-7
- MathSciNet review: 2139931