Geometry of four-folds with three non-commuting involutions

TL;DR

This paper investigates the geometric properties of certain threefolds in a product of projective planes, focusing on their involutions and the dynamics of associated rational maps, extending techniques from K3 surface studies.

Contribution

It adapts techniques from K3 surface geometry to analyze threefolds with three non-commuting involutions and studies their rational self-maps and dynamical degrees.

Findings

Computed the actions of rational maps on divisors.

Determined the first dynamical degrees of map compositions.

Characterized the geometric structure of the threefolds.

Abstract

In this paper we adapt some techniques developed for K3 surfaces, to study the geometry of a family of projective varieties in $\Pl_K^2 \times \Pl_K^2 \times \Pl_K^2$ defined as the intersection of a form of degree $(2,2,2)$ and a form of degree $(1,1,1)$. Members of the family will be equipped with dominant rational self-maps and we will study the actions of those maps on divisors and compute the first dynamical degrees of the composition of any pair.