Moduli of surfaces with an anti-canonical cycle

TL;DR

This paper proves a global Torelli theorem for rational surfaces with anti-canonical cycles and constructs universal families, confirming a conjecture by Friedman from 1984.

Contribution

It establishes a Torelli theorem for pairs (Y,D) with Y a rational surface and D an anti-canonical cycle, and constructs universal families for these pairs.

Findings

Proved a global Torelli theorem for pairs (Y,D).

Constructed natural universal families for such pairs.

Confirmed Friedman's 1984 conjecture.

Abstract

We prove a global Torelli theorem for pairs (Y,D), where Y is a smooth projective rational surface and D is an effective anti-canonical divisor which is a cycle of rational curves. This Torelli theorem was conjectured by Friedman in 1984. In addition, we construct natural universal families for such pairs.