Conics associated with totally degenerate curves

TL;DR

This paper investigates conics arising from totally degenerate stable curves over a field, demonstrating their non-split nature in specific cases and applying these results to determine the period and index of universal genus g curves.

Contribution

It introduces a novel connection between totally degenerate stable curves and associated conics, and applies this to compute period and index for universal genus g curves.

Findings

Conics associated with certain stable curves are non-split.

The period and index of universal genus g curves are both 2g-2 for g ≥ 3.

Abstract

Let $k$ be a field. Let $X/k$ be a stable curve whose geometric irreducible components are smooth rational curves. Taking Stein factorization of its normalization, we get a conic. We show the conic is non-split in certain cases. As an application, we show for $g\geq3$, the period and index of the universal genus $g$ curve both equal to $2g-2$.