Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Picard rank jumps for families of K3 surfaces in positive characteristic

Ruofan Jiang, Ananth N. Shankar, Ziquan Yang

Abstract

Let X/C be a non iso-trivial family of K3 surfaces over a curve C defined over characteristic p > 2 field. We show that if X avoids a necessary and structural obstruction coming from Frobenius, and satisfies a big monodromy condition, then there are infinitely may geometric fibers that have larger Picard rank than the geometric generic fiber.

Picard rank jumps for families of K3 surfaces in positive characteristic

Abstract

Let X/C be a non iso-trivial family of K3 surfaces over a curve C defined over characteristic p > 2 field. We show that if X avoids a necessary and structural obstruction coming from Frobenius, and satisfies a big monodromy condition, then there are infinitely may geometric fibers that have larger Picard rank than the geometric generic fiber.
Paper Structure (59 sections, 70 theorems, 128 equations, 2 figures, 1 table)

This paper contains 59 sections, 70 theorems, 128 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Exceptional families: counterexamples in positive characteristic
  3. Statement of results in the setting of K3 surfaces
  4. Counterexamples with small monodromy
  5. General GSpin Shimura varieties
  6. Outline of proof
  7. Local-global comparison
  8. Supersingular points and decaying lattices of special endomorphisms
  9. Algebraization
  10. Prior and related work
  11. Organization of the paper
  12. Preliminaries on orthogonal Shimura varieties
  13. Basic setup
  14. Newton stratification
  15. Arithmetic intersection theory
  16. ...and 44 more sections

Key Result

Theorem 1.2.1

Let $\mathscr{X}\rightarrow C$ be a non iso-trivial non-exceptional family of K3 surfaces in characteristic $p>2$ such that the discriminant of the generic Picard lattice is prime-to-$p$. Let $L$ be the minimal rank lattice with $\mathscr{X}\rightarrow C$ induced by a map $C\rightarrow \mathscr{S}_L

Figures (2)

  • Figure 1: A picture for the $a_i$'s
  • Figure 2: Line configuration of $\{m_j\cdot \ell_j\}_{1\leq j\leq w}$ with critical points.

Theorems & Definitions (175)

  • Definition 1.1.1
  • Remark 1.1.2
  • Theorem 1.2.1
  • Theorem 1.2.2
  • Corollary 1.2.3
  • Remark 1.2.4
  • Theorem 1.4.1
  • Example 1.5.3
  • Remark 2.1.1
  • Definition 2.1.2
  • ...and 165 more