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Density of Noether-Lefschetz loci for surfaces in Fano and Calabi-Yau threefolds

Edoardo Mason

Abstract

In this paper we provide applications of general results of Baldi-Klingler-Ullmo and Khelifa-Urbanik on the geometry of the Hodge locus associated to an integral polarized variation of Hodge structures to the case of Noether-Lefschetz loci for families of surfaces. In particular, we consider the family of surfaces in the linear system of a sufficiently ample line bundle on a smooth projective threefold $Y$, in case $Y$ is a Fano or a Calabi-Yau threefold, and we discuss the different behaviour of the union of the general, respectively exceptional, components of its Noether-Lefschetz locus.

Density of Noether-Lefschetz loci for surfaces in Fano and Calabi-Yau threefolds

Abstract

In this paper we provide applications of general results of Baldi-Klingler-Ullmo and Khelifa-Urbanik on the geometry of the Hodge locus associated to an integral polarized variation of Hodge structures to the case of Noether-Lefschetz loci for families of surfaces. In particular, we consider the family of surfaces in the linear system of a sufficiently ample line bundle on a smooth projective threefold , in case is a Fano or a Calabi-Yau threefold, and we discuss the different behaviour of the union of the general, respectively exceptional, components of its Noether-Lefschetz locus.

Paper Structure

This paper contains 11 sections, 13 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Context
  3. The density criterion
  4. Noether-Lefschetz loci
  5. Notations
  6. Acknowledgements
  7. Background material
  8. Density of Noether-Lefschetz loci
  9. Other applications
  10. A family of hypersurfaces in $\mathbb{P}^5$
  11. Curves with non-simple Jacobian

Key Result

Theorem 1.2

(khur, Theorem 1.9) Let $(\mathbb{V},\mathscr{F}^{\bullet})$ be a polarized $\mathbb{Z}$-VHS on a smooth connected quasi-projective variety $S$ with generic Hodge datum $(\mathbf{G},D)$. Furthermore, assume that the algebraic monodromy group of $\mathbb{V}$ is $\mathbf{H} = \mathbf{G}^{\text{der}}$ Then the Hodge locus of type $\mathbf{M}$ is analytically dense in $S^{\text{an}}$. Furthermore, if

Theorems & Definitions (28)

  • Conjecture 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • ...and 18 more