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Mathematics > Algebraic Geometry

arXiv:1804.09440 (math)
[Submitted on 25 Apr 2018 (v1), last revised 19 Jun 2018 (this version, v2)]

Title:Density of Noether-Lefschetz loci of polarized irreducible holomorphic symplectic varieties and applications

Authors:Giovanni Mongardi, Gianluca Pacienza
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Abstract:In this note we derive from deep results due to Clozel-Ullmo the density of Noether-Lefschetz loci inside the moduli space of marked (polarized) irreducible holomorphic symplectic (IHS) varieties. In particular we obtain the density of Hilbert schemes of points on projective $K3$ surfaces and of projective generalized Kummer varieties in their moduli spaces. We present applications to the existence of rational curves on projective deformations of such varieties, to the study of relevant cones of divisors, and a refinement of Hassett's result on cubic fourfolds whose Fano variety of lines is isomorphic to a Hilbert scheme of 2 points on a K3 surface. We also discuss Voisin's conjecture on the existence of coisotropic subvarieties on IHS varieties and relate it to a stronger statement on Noether-Lefschetz loci in their moduli spaces.
Comments: v2: Conjectures in section 4 have been modified taking into account the results in 1805.07001, references added, several inaccuracies corrected
Subjects: Algebraic Geometry (math.AG)
MSC classes: 14C99, 14J28, 14J35, 14J40
Cite as: arXiv:1804.09440 [math.AG]
  (or arXiv:1804.09440v2 [math.AG] for this version)
  https://doi.org/10.48550/arXiv.1804.09440

Submission history

From: Giovanni Mongardi [view email]
[v1] Wed, 25 Apr 2018 09:14:29 UTC (34 KB)
[v2] Tue, 19 Jun 2018 09:57:09 UTC (35 KB)
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