Some families of big and stable bundles on K3 surfaces and on their Hilbert schemes of points

2021/09/03 by Gilberto Bini, Bini, Gilberto, Samuel Boissière +3
Mathematics · #14C17 #14D20 #14J28 #14J42 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics

paper · doi:10.48550/arxiv.2109.01598

openalex publication_date 2021/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Here we investigate meaningful families of vector bundles on a very general polarized K3 surface (X,H) and on the corresponding Hyper--Kaehler variety given by the Hilbert scheme of points X^[k]:= \rm Hilb^k(X), for any integer k \geqslant 2. In particular, we prove results concerning bigness and stability of such bundles. First, we give conditions on integers n such that the twist of the tangent bundle of X by the line bundle nH is big and stable on~X; we then prove a similar result for a natural twist of the tangent bundle of X^[k]. Next, we prove global generation, bigness and stability results for tautological bundles on X^[k] arising either from line bundles or from Mukai-Lazarsfeld bundles, as well as from Ulrich bundles on X, using a careful analysis on Segre classes and numerical computations for k = 2, 3.

Related

• Hilbert schemes of points on some K3 surfaces and Gieseker stable bundles
• Special Ulrich bundles on regular surfaces with non-negative Kodaira dimension
• Rigid stable vector bundles on hyperkähler varieties of type K3^[n]
• Hodge numbers of moduli spaces of stable bundles on K3 surfaces
• Stability of Rank 2 Ulrich Bundles on Projective K3 Surfaces