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Moduli of K3 surfaces of degree 2 with four rational double points of type $D_4$

Kazushi Ueda

Abstract

We show that the Satake-Baily-Borel compactification of the moduli space of lattice polarized K3 surfaces parametrizing K3 surfaces of degree 2 with four rational double points of type $D_4$ is the projective 3-space. We also show that the corresponding graded ring of automorphic forms is generated by four elements of weight 2 and one element of weight 11 with one relation of weight 22.

Moduli of K3 surfaces of degree 2 with four rational double points of type $D_4$

Abstract

We show that the Satake-Baily-Borel compactification of the moduli space of lattice polarized K3 surfaces parametrizing K3 surfaces of degree 2 with four rational double points of type is the projective 3-space. We also show that the corresponding graded ring of automorphic forms is generated by four elements of weight 2 and one element of weight 11 with one relation of weight 22.
Paper Structure (10 sections, 7 theorems, 57 equations)

This paper contains 10 sections, 7 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Gorenstein K3 surfaces of degree 2
  3. Double covers of $\mathbb{P}^2$
  4. Lattices and orthogonal groups
  5. Automorphic forms and modular varieties for orthogonal groups
  6. Elliptic K3 surfaces
  7. The complement of $\mathcal{H}_{10}$
  8. The isomorphism ${\overline{\mathcal{M}}} \cong \mathbb{P}^3$
  9. Proof of th:main
  10. K3 surfaces of degree 2 with two rational double points of type $E_8$

Key Result

Lemma 3.1

The space $S_B$ of homogeneous polynomials $f$ of degree 6 in $\mathbb{C}[x_1,x_2,x_3]$, such that the double cover of $\mathbb{P}^2$ branched along the zero of $f$ has singularities which are equal to or worse than rational double points of type $D_4$ above $B$, is a vector space of dimension 4.

Theorems & Definitions (17)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Remark 3.4
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 7 more