The Mori fan of the Dolgachev-Nikulin-Voisin family in genus $2$

TL;DR

The authors deliver a complete combinatorial and geometric description of the Mori fan MF$(\mathcal{Y}/S)$ and the associated secondary fan for the Dolgachev–Nikulin–Voisin family in degree $2$, yielding a precise enumeration of all maximal cones and birational orbits. Central to the work is a detailed analysis of degree two degenerations of $K3$ surfaces in $(-1)$-form, encoded by two sphere triangulations $\mathscr P$ and $\mathscr T$, which determine models of class $\mathscr P$ and class $\mathscr T$ respectively. They prove MF$(\mathcal{Y}/S)$ has $3460$ maximal cones, distributed as $753$ from class $\mathscr T$ and $2707$ from class $\mathscr P$, with $588$ orbits under $\operatorname{Bir}(\mathcal{Y}/S)$ (comprising $457$ orbits of class $\mathscr P$ models and $131$ orbits of class $\mathscr T$ models). The secondary fan is shown to have four maximal cones and two birational orbits, and is a coarser invariant than the Mori fan. Overall, the results substantiate the GHKS program in the degree-two case by providing an explicit, finite combinatorial framework for compactifications of moduli spaces of polarized $K3$ surfaces via MMP and mirror-symmetric degeneration theory.

Abstract

In this paper we study the Mori fan of the Dolgachev-Nikulin-Voisin family in degree $2$ as well as the associated secondary fan. The main result is an enumeration of all maximal dimensional cones of the two fans.

Figures (18)

• Figure 1: An elementary modification of type I
• Figure 2: An elementary modification of type II
• Figure 3: The refinement of a triangle of $\Gamma'$.
• Figure 8: The triangulations $\mathscr{P}$ and $\mathscr{T}$.
• Figure 9: The triangulations with $4$ triangles.

Theorems & Definitions (215)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1: cf. Frie83
• Definition 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Definition 2.6
• Theorem 2.7: cf. MiMo
• Definition 2.8