Deformation and Unobstructedness of Determinantal Schemes
| Autor: | Jan Kleppe, Rosa Miró-Roig |
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| Rok vydání: | 2023 |
| Předmět: | |
| Zdroj: | Memoirs of the American Mathematical Society. 286 |
| ISSN: | 1947-6221 0065-9266 |
| Popis: | A closed subscheme X ⊂ P n X\subset \mathbb {P}^n is said to be determinantal if its homogeneous saturated ideal can be generated by the s × s s\times s minors of a homogeneous p × q p\times q matrix satisfying ( p − s + 1 ) ( q − s + 1 ) = n − dim X (p-s+1)(q-s+1)=n - \dim X and it is said to be standard determinantal if, in addition, s = min ( p , q ) s=\min (p,q) . Given integers a 1 ≤ a 2 ≤ ⋯ ≤ a t + c − 1 a_1\le a_2\le \cdots \le a_{t+c-1} and b 1 ≤ b 2 ≤ ⋯ ≤ b t b_1\le b_2 \le \cdots \le b_t we consider t × ( t + c − 1 ) t\times (t+c-1) matrices A = ( f i j ) \mathcal {A}=(f_{ij}) with entries homogeneous forms of degree a j − b i a_j-b_i and we denote by W ( b _ ; a _ ; r ) ¯ \overline {W(\underline {b};\underline {a};r)} the closure of the locus W ( b _ ; a _ ; r ) ⊂ H i l b p ( t ) ( P n ) W(\underline {b};\underline {a};r)\subset Hilb^{p(t)}(\mathbb {P}^{n}) of determinantal schemes defined by the vanishing of the ( t − r + 1 ) × ( t − r + 1 ) (t-r+1)\times (t- r+1) minors of such A \mathcal {A} for max { 1 , 2 − c } ≤ r > t \max \{1,2-c\} \le r > t . W ( b _ ; a _ ; r ) W(\underline {b};\underline {a};r) is an irreducible algebraic set. First of all, we compute an upper r r -independent bound for the dimension of W ( b _ ; a _ ; r ) W(\underline {b};\underline {a};r) in terms of a j a_j and b i b_i which is sharp for r = 1 r=1 . In the linear case ( a j = 1 , b i = 0 a_j = 1, b_i=0 ) and cases sufficiently close, we conjecture and to a certain degree prove that this bound is achieved for all r r . Then, we study to what extent the family W ( b _ ; a _ ; r ) W(\underline {b};\underline {a};r) fills in a generically smooth open subset of the corresponding component of the Hilbert scheme H i l b p ( t ) ( P n ) Hilb^{p(t)}(\mathbb {P}^{n}) of closed subschemes of P n \mathbb {P}^n with Hilbert polynomial p ( t ) ∈ Q [ t ] p(t)\in \mathbb {Q}[t] . Under some weak numerical assumptions on the integers a j a_j and b i b_i (or under some depth conditions) we conjecture and often prove that W ( b _ ; a _ ; r ) ¯ \overline {W(\underline {b};\underline {a};r)} is a generically smooth component. Moreover, we also study the depth of the normal module of the homogeneous coordinate ring of ( X ) ∈ W ( b _ ; a _ ; r ) (X)\in W(\underline {b};\underline {a};r) and of a closely related module. We conjecture, and in some cases prove, that their codepth is often 1 (resp. r 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 ) W(\underline {b};\underline {a};1) to W ( b _ ; a _ ; r ) W(\underline {b};\underline {a};r) with 1 ≤ r > t 1\le r > t and c ≥ 2 − r c\ge 2-r . Finally, deformations of exterior powers of the cokernel of the map determined by A \mathcal {A} are studied and proven to be given as deformations of X ⊂ P n X \subset \mathbb {P}^n if dim X ≥ 3 \dim X \ge 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. |
| Databáze: | OpenAIRE |
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