New derived autoequivalences of Hilbert schemes and generalised Kummer varieties

2013/01/21 by Andreas Krug, Krug, Andreas
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1301.4970

openalex publication_date 2013/01/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that for every smooth projective surface X and every n≥ 2 the push-forward along the diagonal embedding gives a P^n-1-functor into the S_n-equivariant derived category of Xⁿ. Using the Bridgeland--King--Reid--Haiman equivalence this yields a new autoequivalence of the derived category of the Hilbert scheme of n points on X. In the case that the canonical bundle of X is trivial and n=2 this autoequivalence coincides with the known EZ-spherical twist induced by the boundary of the Hilbert scheme. We also generalise the 16 spherical objects on the Kummer surface given by the exceptional curves to n⁴ orthogonal P^n-1-Objects on the generalised Kummer variety.

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