The moduli space of rational elliptic surfaces of index two

TL;DR

This paper constructs a nine-dimensional moduli space for marked rational elliptic surfaces of index two, including its compactifications as quotients of affine space, advancing understanding of their geometric classification.

Contribution

It introduces a new moduli space for marked rational elliptic surfaces of index two as a non-complete toric variety and provides explicit compactifications via algebraic quotients.

Findings

Constructed a 9-dimensional moduli space as a non-complete toric variety.

Developed compactifications as quotients of affine space by algebraic group actions.

Enhanced the geometric understanding of rational elliptic surfaces of index two.

Abstract

In this paper we construct a moduli space for marked rational elliptic surfaces of index two as a non-complete toric variety of dimension nine. We also construct compactifications of this moduli space, which are obtained as quotients of $\mathbb{A}^{12}$ by an action of $\mathbb{G}_m^3$.