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K3 surfaces over $\mathbb{Q}$ of degree $10$ that have Picard rank $1$

Victor de Vries

Abstract

We give examples of K3 surfaces over $\mathbb{Q}$ of degree $10$ whose geometric Picard group has rank~$1$. These K3 surfaces are intersections in $\mathbb{P}^9$ of three hyperplanes, one quadric and the image of the Plücker embedding of the Grasmannian $\mathrm{Gr}(2,5)$. We also give an example of a K3 surface of degree $6$ over~$\mathbb{Q}$ whose Picard rank is $1$.

K3 surfaces over $\mathbb{Q}$ of degree $10$ that have Picard rank $1$

Abstract

We give examples of K3 surfaces over of degree whose geometric Picard group has rank~. These K3 surfaces are intersections in of three hyperplanes, one quadric and the image of the Plücker embedding of the Grasmannian . We also give an example of a K3 surface of degree over~ whose Picard rank is .
Paper Structure (4 sections, 19 theorems, 19 equations)

This paper contains 4 sections, 19 theorems, 19 equations.

Table of Contents

  1. Preliminaries
  2. The surfaces $S_2$ and $S_3$
  3. The degree $10$ example
  4. The degree $6$ example

Key Result

Theorem 1

Let $S$ be a subscheme of $\mathrm{Gr}(2,5)_{\mathbb{Q}}$ that is the zero locus of forms $L_1,L_2,L_3,Q$, where the $L_i$ are lifts of $l_i,l_i'$ to forms over $\mathbb{Z}$ and $Q$ is a lift of $q,q'$ to a form over $\mathbb{Z}$. The scheme $S$ is a K3 surface over $\mathbb{Q}$ of degree $10$, whos

Theorems & Definitions (42)

  • Theorem 1
  • Lemma 1.1
  • proof
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Lemma 1.5
  • Remark 1.6
  • Lemma 1.7
  • proof
  • ...and 32 more