The degeneration of a family of rational surface automorphisms

TL;DR

This paper studies a family of rational surface automorphisms, focusing on their degeneration behavior and the induced birational maps with specific dynamical degrees.

Contribution

It demonstrates how to resolve indeterminacies in degenerations of rational surface automorphisms and computes the dynamical degree of the induced map.

Findings

The limiting map in the degeneration is the identity on the special fiber.

Blowing up at an indeterminate curve yields a birational map on the exceptional divisor.

The induced birational map has a dynamical degree of 16.

Abstract

We consider a one-dimensional family of rational surfaces with automorphisms. In a degeneration of this family, the limiting map is the identity map on a special fiber. We check that the map on the total space of the family has indeterminacy in the special fiber. However, we show that after blowing-up at an indeterminate curve, there is an induced birational map on the exceptional divisor over the indeterminate curve. Moreover, we show that this map has dynamical degree 16.