Geometric interpretation of toroidal compactifications of moduli of points in the line and cubic surfaces

TL;DR

This paper explores the geometric structure of certain compactifications of moduli spaces, revealing their equivalence to moduli of stable pairs and providing a Hodge-theoretic interpretation, advancing understanding in algebraic geometry.

Contribution

It establishes isomorphisms between toroidal compactifications and moduli spaces of stable pairs, linking geometric and Hodge-theoretic perspectives.

Findings

Toroidal compactifications are isomorphic to moduli of stable pairs.

Provides a Hodge-theoretic interpretation for eight points in the projective line.

Connects GIT, Baily-Borel, and MMP frameworks in moduli theory.

Abstract

It is known that some GIT compactifications associated to moduli spaces of either points in the projective line or cubic surfaces are isomorphic to Baily-Borel compactifications of appropriate ball quotients. In this paper, we show that their respective toroidal compactifications are isomorphic to moduli spaces of stable pairs as defined in the context of the MMP. Moreover, we give a precise mixed-Hodge-theoretic interpretation of this isomorphism for the case of eight labeled points in the projective line.