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I-surfaces from surfaces with one exceptional unimodal point

Sönke Rollenske, Diana Torres

Abstract

We complement recent work of Gallardo, Pearlstein, Schaffler, and Zhang, showing that the stable surfaces with $K_X^2 =1$ and $χ(\mathcal O_X) = 3$ they construct are indeed the only ones arising from imposing an exceptional unimodal double point. In addition, we explicitly describe the birational type of the surfaces constructed from singularities of type $E_{12}$, $E_{13}$, and $E_{14}$.

I-surfaces from surfaces with one exceptional unimodal point

Abstract

We complement recent work of Gallardo, Pearlstein, Schaffler, and Zhang, showing that the stable surfaces with and they construct are indeed the only ones arising from imposing an exceptional unimodal double point. In addition, we explicitly describe the birational type of the surfaces constructed from singularities of type , , and .
Paper Structure (7 sections, 17 theorems, 52 equations, 4 figures, 6 tables)

This paper contains 7 sections, 17 theorems, 52 equations, 4 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Gorenstein surfaces with one elliptic Gorenstein singularity
  3. Surfaces with exceptional unimodal double points of type $E_n$
  4. Proof of Theorem \ref{['thm: type E']}
  5. Surfaces with an exceptional unimodal double points of type $Z_n$ or $W_n$
  6. Proof of Theorem \ref{['thm: Z and W']}
  7. Application to the moduli space of stable surfaces $\overline{\mathfrak M}_{1,3}$

Key Result

Lemma 2.5

In the above situation we have the following:

Figures (4)

  • Figure 1: Schematic construction of surfaces with $E_n$ type singularities
  • Figure 2: Schematic construction, type $E_{12}$ (picture)
  • Figure 3: Construction of surfaces $W$ with one singularity of type $Z_n$ or $W_n$
  • Figure :

Theorems & Definitions (41)

  • Example 2.1
  • Example 2.2
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • Remark 3.1
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • ...and 31 more