Nets of quadric surfaces and plane cubics and their GIT stability

TL;DR

This paper explores the geometric and stability properties of nets of quadrics, plane cubics, and associated quartic curves, establishing criteria for GIT stability using birational geometry.

Contribution

It generalizes the cycle of correspondences from nets of quadrics to rational elliptic threefolds and provides a complete GIT stability criterion for these objects.

Findings

Established a criterion for GIT stability of nets of quadrics, plane cubics, and quartic curves.

Connected nets of quadrics with rational elliptic threefolds through birational geometry.

Analyzed the geometric relations and stability conditions of these algebraic objects.

Abstract

A general net of quadric surfaces, together with a choice of a base point, defines a net of plane cubics via the Gale transformation of the remaining seven base points. To both nets, one can also naturally associate the same smooth plane quartic. In this paper, we generalize the cycle of correspondences arising from nets of quadrics that define rational elliptic threefolds and provide a complete criterion for GIT stability of the three underlying geometric objects using birational-geometric techniques.