Summary
Let X be an algebraic K3 surface with Picard lattice N(X), and M X (v) the moduli space of sheaves on X with given primitive isotropic Mukai vector v = (r, H, s). In [14] and [3], we described all the divisors in moduli of polarized K3 surfaces (X, H) (that is, all pairs H ∈ N(X) with rank N(X) = 2) for which M X (v)≅X. These provide certain Mukai self-correspondences of X. Applying these results, we show that there exists a Mukai vector v and a codimension-2 subspace in moduli of (X, H) (that is, a pair H ∈ N(X) with rank N(X) = 3) for which M X (v)≅X, but such that this subspace does not extend to a divisor in moduli having the same property. There are many similar examples. Aiming to generalize the results of [14] and [3], we discuss the general problem of describing all subspaces of moduli of K3 surfaces with this property, and the Mukai self-correspondences defined by these and their composites, in an attempt to outline a possible general theory.
2000 Mathematics Subject Classifications: 14D20, 14J28, 14J60, 14J10, 14C25, 11E12
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To Yuri Ivanovich Manin for his 70th birthday
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Nikulin, V.V. (2010). Self-Correspondences of K3 Surfaces via Moduli of Sheaves. In: Tschinkel, Y., Zarhin, Y. (eds) Algebra, Arithmetic, and Geometry. Progress in Mathematics, vol 270. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4747-6_14
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