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Deformations of I-surfaces with elliptic singularities

Robert Friedman, Phillip Griffiths

Abstract

An I-surface $S$ is an algebraic surface of general type with $K_S^2 = 1$ and $p_g(S) = 2$. Recent research has centered on trying to give an explicit description of the KSBA compactification of the moduli space of these surfaces. The possible normal Gorenstein examples have been enumerated by work of Franciosi-Pardini-Rollenske. The goal of this paper is to give a more precise description of such surfaces in case their singularities are simple elliptic and/or cusp singularities, and to work out their deformation theory. In particular, under some mild general position assumptions, we show that deformations of the surfaces in question are versal for deformations of the singular points, with two exceptions where the discrepancy is analyzed in detail.

Deformations of I-surfaces with elliptic singularities

Abstract

An I-surface is an algebraic surface of general type with and . Recent research has centered on trying to give an explicit description of the KSBA compactification of the moduli space of these surfaces. The possible normal Gorenstein examples have been enumerated by work of Franciosi-Pardini-Rollenske. The goal of this paper is to give a more precise description of such surfaces in case their singularities are simple elliptic and/or cusp singularities, and to work out their deformation theory. In particular, under some mild general position assumptions, we show that deformations of the surfaces in question are versal for deformations of the singular points, with two exceptions where the discrepancy is analyzed in detail.
Paper Structure (35 sections, 62 theorems, 184 equations)

This paper contains 35 sections, 62 theorems, 184 equations.

Table of Contents

  1. Deformations of simple elliptic and cusp singularities
  2. Preliminary results on simple elliptic and cusp singularities
  3. Adjacencies of some elliptic singularities
  4. Deformations of elliptic singularities: the local case
  5. Deformations of elliptic singularities: the global case
  6. Some variations of the above
  7. Deformations of I-surfaces are unobstructed
  8. Structure of I-surfaces with elliptic singularities
  9. Some preliminary results
  10. Numerology of I-surfaces
  11. The main geometric result
  12. The case $\kappa(\widetilde{Y}) =1$
  13. The cotangent bundle
  14. Multiple fibers of multiplicity $2$ and double covers
  15. A vanishing theorem: the case where $D$ is irreducible
  16. ...and 20 more sections

Key Result

Theorem 2

Let $Y$ be a normal Gorenstein surface with at worst rational double point, simple elliptic and cusp singularities such that $\omega_Y$ is ample, $\omega_Y^2=1$, and $\dim H^0(Y; \omega_Y) = 2$. Then $Y$ is isomorphic to a weighted hypersurface of degree $10$ contained in the smooth locus of the wei

Theorems & Definitions (146)

  • Definition 1
  • Theorem 2
  • Theorem 3
  • Definition 1.1
  • Remark 1.2
  • Lemma 1.3
  • Definition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Lemma 1.7
  • ...and 136 more