Deformation and unobstructedness of determinantal schemes / Rosa M. Miró-Roig, Jan O. Kleppe.

Miró-Roig, Rosa M. [Browse]
  • Providence, RI : American Mathematical Society, 2023.
  • ©2023
v, 113 pages : illustrations ; 26 cm


Summary note
A closed subscheme ... is said to be determinantal if its homogeneous saturated ideal can be generated by the ... minors of a homogeneous ... matrix satisfying ... and it said to be standard determinantal if, in addition, ... Given integers ... and ... we consider ... with entries homogeneous forms of degree ... and we denote by ... the closer of the locus ... of determinantal schemes define by the vanishing of the ... minors of such ... for ... is an irreducible algebraic set. First of all, we compute an upper r-independent bound the dimension of ... in terms of ... and ... which is sharp for ... In the linear case ... 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 ... fills in a generally smooth open subset of the corresponding component of the Hilbert scheme ... of close subschemes of ... with Hilbert polynomial ... Under some weak numerical assumptions on the integers ... (or under some depth conditions) we conjecture and often prove that ... is a generically smooth component. Moreover, we also study the depth of the normal module of the homogeneous coordinate ring of ... 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 ... to ... with ... Finally, deformations of exterior powers of the cokernel of the map determined by ... are studied and proven to be given as deformations of ... if dim ... The work contains many example which illustrate the results obtained and a considerable number of open problems; some of them are collected as conjectures in the final section.
"June 2023, volume 286, number 1418 (second of 6 numbers)."
Bibliographic references
Includes bibliographical references (pages 111-113).
  • Chapter 1. Introduction
  • Chapter 2. Preliminaries
  • Chapter 3. Families of standard determinantal scheme
  • Chapter 4. Unobstructedness of quotients of zerosections
  • Chapter 5. Deformation of minors
  • Chapter 6. The dimension of the determinantal locus
  • Chapter 7. Generically smooth components of the Hilbert scheme
  • Chapter 8. Computing dimensions by deleting columns
  • Chapter 9. Deformations of exterior powers of modules over determinantal schemes
  • Chapter 10. Final comments and conjectures
  • Bibliography.
  • 9781470463113
  • 1470463113
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