Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Combinatorial K3 surfaces and the Mori fan of the Dolgachev--Nikulin--Voisin family in degree 2

Klaus Hulek, Christian Lehn

TL;DR

The paper develops a combinatorial framework to study the Mori fan of degree-2 Dolgachev–Nikulin–Voisin degenerations by replacing the ambient threefold with combinatorial K3 surfaces determined by their central fibers. It introduces a combinatorial pushforward φ_c on Picard groups, tied to type I and II flops, and shows this map captures the action of the geometric flop on Pic(Y) via restriction from the normalization. By expressing the cone of curves and nef cone in terms of curve-structure data, the work yields explicit, computable descriptions of the Mori fan and enables algorithmic implementation. The degree-2 setting yields a concrete, tractable model with seeds Y_P and Y_T and a detailed account of how flops transform curve structures, interior special points, and gluings, providing both theoretical insight and a path to practical computation.

Abstract

We introduce the notion of a combinatorial K3 surface. Those form a certain class of type III semistable K3 surfaces and are completely determined by combinatorial data called curve structures. Emphasis is put on degree $2$ combinatorial K3 surfaces, but the approach can be used to study higher degree as well. We describe elementary modifications both in terms of the curve structures as well as on the Picard groups. Together with a description of the nef cone in terms of curve structures, this provides an approach to explicitly computing the Mori fan of the Dolgachev--Nikulin--Voisin family in degree $2$.

Combinatorial K3 surfaces and the Mori fan of the Dolgachev--Nikulin--Voisin family in degree 2

TL;DR

The paper develops a combinatorial framework to study the Mori fan of degree-2 Dolgachev–Nikulin–Voisin degenerations by replacing the ambient threefold with combinatorial K3 surfaces determined by their central fibers. It introduces a combinatorial pushforward φ_c on Picard groups, tied to type I and II flops, and shows this map captures the action of the geometric flop on Pic(Y) via restriction from the normalization. By expressing the cone of curves and nef cone in terms of curve-structure data, the work yields explicit, computable descriptions of the Mori fan and enables algorithmic implementation. The degree-2 setting yields a concrete, tractable model with seeds Y_P and Y_T and a detailed account of how flops transform curve structures, interior special points, and gluings, providing both theoretical insight and a path to practical computation.

Abstract

We introduce the notion of a combinatorial K3 surface. Those form a certain class of type III semistable K3 surfaces and are completely determined by combinatorial data called curve structures. Emphasis is put on degree combinatorial K3 surfaces, but the approach can be used to study higher degree as well. We describe elementary modifications both in terms of the curve structures as well as on the Picard groups. Together with a description of the nef cone in terms of curve structures, this provides an approach to explicitly computing the Mori fan of the Dolgachev--Nikulin--Voisin family in degree .
Paper Structure (20 sections, 21 theorems, 30 equations, 12 figures)

This paper contains 20 sections, 21 theorems, 30 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Combinatorial K3 surfaces
  3. Models of the DNV family
  4. Type I flops
  5. Combinatorial K3 surfaces of degree 2
  6. Curve structures
  7. Interior special points
  8. Rulings on combinatorial K3 surfaces of class T
  9. Rulings on combinatorial K3 surfaces of class P
  10. Type II flops
  11. Type II flops of combinatorial K3 surfaces of degree 2
  12. Type II flops on curve structures
  13. Bases and cones
  14. Type I flops on curve structures
  15. Curves on combinatorial K3 surfaces
  16. ...and 5 more sections

Key Result

Proposition 2.7

Taking the dual intersection complex defines a bijection between the set of isomorphism classes of combinatorial K3 surfaces in $(-1)$-form and triangulations of the sphere ${\mathbb S}^2$ such that no vertex has valency greater than $6$. In particular, a combinatorial K3 surface in $(-1)$-form is u

Figures (12)

  • Figure 1: A type I flop (on the normalizations) of combinatorial K3 surfaces.
  • Figure 2: The triangulations $P$ and $T$ of ${\mathbb S}^2$.
  • Figure 3: The augmented curve structure of the surface ${\mathfrak Y}_1$.
  • Figure 4: The augmented curve structure of the surface ${\mathfrak Y}_2$.
  • Figure 5: The augmented curve structure of the surface ${\mathfrak Y}_4$.
  • ...and 7 more figures

Theorems & Definitions (78)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.6
  • Proposition 2.7
  • Definition 2.8
  • Remark 2.9
  • Definition 2.12
  • Remark 2.13
  • ...and 68 more