On the ample cone of a rational surface with an anticanonical cycle

TL;DR

This paper explores the geometry of rational surfaces with anticanonical cycles, generalizing Looijenga's results to cases with more than five components, and examines how certain isometries relate to the ample cone and specific divisor classes.

Contribution

It extends Looijenga's analysis of the ample cone and divisor classes to more complex anticanonical cycles, connecting isometries with geometric structures in these surfaces.

Findings

Relationship between isometries and the ample cone established

Conditions for preserving the set R analyzed

Generalization of Looijenga's results to more components

Abstract

Let $Y$ be a smooth rational surface and let $D$ be a cycle of rational curves on $Y$ which is an anticanonical divisor, i.e. an element of $|-K_Y|$. Looijenga studied the geometry of such surfaces $Y$ in case $D$ has at most five components and identified a geometrically significant subset $R$ of the divisor classes of square -2 orthogonal to the components of $D$. Motivated by recent work of Gross, Hacking, and Keel on the global Torelli theorem for pairs $(Y,D)$, we attempt to generalize some of Looijenga's results in case $D$ has more than five components. In particular, given an integral isometry $f$ of $H^2(Y)$ which preserves the classes of the components of $D$, we investigate the relationship between the condition that $f$ preserves the "generic" ample cone of $Y$ and the condition that $f$ preserves the set $R$.