The moduli space of rational elliptic surfaces of index two (Q2166095)

If \(Y\) is a smooth projective rational surface which admits a relatively minimal fibration \(\mathcal{E} : Y \to \mathbb{P}^{1}\) of genus 1, there exists an integer \(m \geq 1\) such that \(\mathcal{E}\) realizes the anti-pluricanonical system \(|-mK_{Y}|\), and \(\mathcal{E}\) admits a rational \(m-\)section \(\theta.\) The fiber components that do not meet \(\theta\) can all be contracted to obtain an elliptic surface \(\overline{Y}\) with at worst rational double point singularities, and \(\theta\) is relatively ample for the resulting fibration \(\overline{\pi} : \overline{Y} \to \mathbb{P}^{1}\) The pair \((\overline{Y},\theta)\) is called a marked RES (rational elliptic surface) of index \(m.\) Marked RES's of index \(1\) and unmarked RES's of index \(2\) were previously studied by the first author in [Math. Ann. 255, 379--394 (1981; Zbl 0438.14023)] and by the second author in [``Stability of pencils of plane sextics and Halphen pencils of index two,'' Preprint, \url{arXiv:2101.03152}], respectively. The main result of the paper under review is that the moduli space of marked RES's of index 2 exists as a non-complete toric variety of dimension \(9,\) namely a quotient of a single affine toric variety by \(\mathbb{G}_{m},\) and that it admits complete toric compactifications which are obtained as quotients of \(\mathbb{A}^{12}\) by an action of \(\mathbb{G}^{3}_{m}.\) The authors first show that the data of a marked RES \((\overline{Y},\theta)\) of index 2 is equivalent to a curve in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\) of bidegree \((4,3)\) having a double point tacnode and at worst ADE singularities, as well as some additional properties; this curve arises as the branch locus of a double cover associated to \((\overline{Y},\theta).\) Starting with a normal form for such branch loci, they extract the \(\mathbb{G}_{m}^{3}-\)action on \(\mathbb{A}^{12}\) referred to in the second part of the main result, which is used in two ways. One is to obtain a further normalization that yields the existence of the moduli space claimed in the first part of the main result. The other is to analyze the wall-crossing behavior of its GIT quotients and identify the chambers that give complete toric compactifications of the moduli space. At the end of the paper, the authors investigate the quotients that arise from an adjacent pair of these chambers and describe the surfaces parametrized by their boundaries.