Notes on diagonals of the product and symmetric variety of a surface

JOURNAL OF PURE AND APPLIED ALGEBRA

卷:224

Article

Scala, Luca^1,2 

[1] Puc Rio, Dept Matemat, Rua Marques Sao Vicente 225, BR-22451900 Gavea, RJ, Brazil

[2] Univ Fed Rio de Janeiro, Inst Matemat, Ctr Technol, Av Athos da Silveira Ramos 149, BR-21941909 Rio De Janeiro, RJ, Brazil

关键词: Diagonal ideals;    Hilbert scheme of points;    Determinant line bundles;    Regularity;   

DOI  :  10.1016/j.jpaa.2020.106352

来源: Elsevier

【 摘 要 】

Let X be a smooth quasi-projective algebraic surface and let Delta(n) be the big diagonal in the product variety X-n. We study cohomological properties of the ideal sheave I-Delta n(k) their invariants (I-Delta n(k)) S-n by the symmetric group, seen as ideal sheaves over the symmetric variety (SX)-X-n. In particular we obtain resolutions of the sheaves of invariants (I-Delta n) S-n for n = 3, 4 in terms of invariants of sheaves over X-n whose cohomology is easy to calculate. Moreover, we relate, via the Bridgeland-King-Reid equivalence, powers of determinant line bundles over the Hilbert scheme to powers of ideals of the big diagonal Delta(n). We deduce applications to the cohomology of double powers of determinant line bundles over the Hilbert scheme with 3 and 4 points and we give universal formulas for their Euler-Poincare characteristic. Finally, we obtain upper bounds for the regularity of the sheaves I(Delta n)(k )over X-n with respect to very ample line bundles of the form L boxed times ... boxed times L and of their sheaves of invariants )en on the symmetric variety (SX)-X-n with respect to very ample line bundles of the form D-L. (C) 2020 Elsevier B.V. All rights reserved.

【 授权许可】

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