Deformation and Unobstructedness of Determinantal Schemes / Jan O. Kleppe and Rosa M. Miró-Roig.

Format:

Book

Author/Creator:

Kleppe, Jan O., 1947- author.

Miró-Roig, Rosa M., author.

Series:

Memoirs of the American Mathematical Society ; Volume 286.

Memoirs of the American Mathematical Society Series ; Volume 286

Language:

English

Subjects (All):

Schemes (Algebraic geometry).

Surfaces, Deformation of.

Determinantal varieties.

Physical Description:

1 online resource (126 pages)

Edition:

First edition.

Place of Publication:

Providence, RI : American Mathematical Society, [2023]

Summary:

"A closed subscheme X [is a proper subset of] Pn is said to be determinantal if its homogeneous saturated ideal can be generated by the s [times] s minors of a homogeneous p [times] q matrix satisfying (p [minus] s [plus] 1)(q [minus] s [plus] 1) [equals] n [minus] dimX and it is said to be standard determinantal if, in addition, s [equals] min(p, q). Given integers a1 [leq] a2 [leq] [dots] [leq] at[plus]c[minus]1 and b1 [leq] b2 [leq] [dots] [leq] bt we consider t [times] (t [plus] c [minus] 1) matrices A [equals] (fij) with entries homogeneous forms of degree aj [minus] bi and we denote by W(b; a; r) the closure of the locus W(b; a; r) [is a proper subset of] Hilbp(t)(Pn) of determinantal schemes defined by the vanishing of the (t[minus]r[plus]1)[times](t[minus]r[plus]1) minors of such A for max{1, 2[minus]c} [leq] r [less than] t. W(b; a; r) is an irreducible algebraic set. First of all, we compute an upper r-independent bound for the dimension of W(b; a; r) in terms of aj and bi which is sharp for r [equals] 1. In the linear case (aj [equals] 1, bi [equals] 0) and cases sufficiently close, we conjecture and to a certain degree prove that this bound is achieved for all r. Then, we study to what extent the family W(b; a; r) fills in a generically smooth open subset of the corresponding component of the Hilbert scheme Hilbp(t)(Pn) of closed subschemes of Pn with Hilbert polynomial p(t) [an element of] Q[t]. Under some weak numerical assumptions on the integers aj and bi (or under some depth conditions) we conjecture and often prove that W(b; a; r) is a generically smooth component. Moreover, we also study the depth of the normal module of the homogeneous coordinate ring of (X) [an element of] W(b; a; r) and of a closely related module. We conjecture, and in some cases prove, that their codepth is often 1 (resp. r). These results extend previous results on standard determinantal schemes to determinantal schemes; i.e. previous results of the authors on W(b; a; 1) to W(b; a; r) with 1 [leq] r [less than] t and c [geq] 2 [minus] r. Finally, deformations of exterior powers of the cokernel of the map determined by A are studied and proven to be given as deformations of X [is a proper subset of] Pn if dimX [geq] 3. The work contains many examples which illustrate the results obtained and a considerable number of open problems; some of them are collected as conjectures in the final section"-- Provided by publisher.

Contents:

Preliminaries

Families of standard determinantal schemes

Unobstructedness of quotients of zerosections

Deformation of minors

The dimension of the determinantal locus

Generically smooth components of the Hilbert scheme

Computing dimensions by deleting columns

Deformations of exterior powers of modules over determinantal

schemes

Final comments and conjectures.

Notes:

Description based on print version record.

Includes bibliographical references.

Other Format:

Print version: Kleppe, Jan O. Deformation and Unobstructedness of Determinantal Schemes

ISBN:

9781470475123

147047512X