Deformation and unobstructedness of determinantal schemes / Jan O. Kleppe, Rosa M. Miro-Roig.
| Author/creator | Kleppe, Jan O., 1947- |
| Other author | Miró-Roig, Rosa M. |
| Format | Electronic |
| Publication Info | Providence, RI : American Mathematical Society, 2023. |
| Description | v, 113 pages : illustrations ; 26 cm. |
| Supplemental Content | Full text available from Ebook Central - Academic Complete |
| Subjects |
| Series | Memoirs of the American Mathematical Society, 0065-9266 ; Volume 286, number 1418 |
| 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. |
| Abstract | "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. |
| Bibliography note | Includes bibliographical references (pages 111-113). |
| Access restriction | Available only to authorized users. |
| Technical details | Mode of access: World Wide Web |
| Genre/form | Electronic books. |
| LCCN | 2023031259 |
| ISBN | 9781470463113 (paperback) |
| ISBN | (pdf) |
Availability
| Library | Location | Call Number | Status | Item Actions |
|---|---|---|---|---|
| Electronic Resources | Access Content Online | ✔ Available |