Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

"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"--

Schemes (Algebraic geometry) --- Surfaces, Deformation of. --- Determinantal varieties.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Mathematical statistics --- Geometry, Algebraic --- Algebraic geometry -- Instructional exposition (textbooks, tutorial papers, etc.) --- Algebraic geometry -- Real algebraic and real analytic geometry -- Semialgebraic sets and related spaces. --- Algebraic geometry -- Special varieties -- Determinantal varieties. --- Algebraic geometry -- Special varieties -- Toric varieties, Newton polyhedra. --- Algebraic geometry -- Tropical geometry -- Tropical geometry. --- Biology and other natural sciences -- Genetics and population dynamics -- Problems related to evolution. --- Commutative algebra -- Computational aspects and applications -- GrÃjabner bases; other bases for ideals and modules (e.g., Janet and border bases) --- Convex and discrete geometry -- Polytopes and polyhedra -- Lattice polytopes (including relations with commutative algebra and algebraic geometry) --- Operations research, mathematical programming -- Mathematical programming -- Integer programming. --- Probability theory and stochastic processes -- Markov processes -- Markov chains (discrete-time Markov processes on discrete state spaces) --- Statistics -- Instructional exposition (textbooks, tutorial papers, etc.) --- Statistics -- Multivariate analysis -- Contingency tables. --- Statistics -- Parametric inference -- Hypothesis testing. --- Commutative algebra -- Computational aspects and applications -- Solving polynomial systems; resultants.