Singularities of the isospectral Hilbert scheme

Scala, Luca

doi:10.1080/00927872.2019.1570226

Cited by 1 publication

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“…The main reason for studying diagonal ideals is that their geometry is tightly intertwined with the geometry of the Hilbert scheme of points X [n] and that of the isospectral Hilbert scheme B n . As an example of this close interplay, we mention that in [Sca15b] we related the singularities of the isospectral Hilbert scheme in terms of the singularities of the pair (X n , I ∆n ) and it is by studying the latter that we could prove that the singularities of B n are canonical if n ≤ 5, log-canonical if n ≤ 7 and not log-canonical if n ≥ 9.…”

Section: Introductionmentioning

confidence: 99%

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

Scala

2020

Journal of Pure and Applied Algebra

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Let X be a smooth quasi-projective algebraic surface and let ∆n the big diagonal in the product variety X n . We study cohomological properties of the ideal sheaves I k ∆n and their invariants (I k ∆n ) Sn by the symmetric group, seen as ideal sheaves over the symmetric variety S n X. In particular we obtain resolutions of the sheaves of invariants (I∆ n ) Sn 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 ∆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-Poincaré characteristic. Finally, we obtain upper bounds for the regularity of the sheaves I k ∆n over X n with respect to very ample line bundles of the form L ⊠ · · · ⊠ L and of their sheaves of invariants (I k ∆n ) Sn on the symmetric variety S n X with respect to very ample line bundles of the form DL.

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Section: Introductionmentioning

confidence: 99%