The Torelli map restricted to the hyperelliptic locus

TL;DR

This paper investigates the behavior of the Torelli map on hyperelliptic curves, showing it is an immersion in characteristic not 2, but fails to be so in characteristic 2 due to inseparability issues.

Contribution

It establishes the conditions under which the Torelli map restricted to hyperelliptic locus is an immersion, highlighting differences between characteristics.

Findings

Immersion in characteristic not 2

Failure of immersion in characteristic 2

Kernel of tangent map has dimension g-2 in characteristic 2

Abstract

Let $g \geq 2$ and let the Torelli map denote the map sending a genus $g$ curve to its principally polarized Jacobian. We show that the restriction of the Torelli map to the hyperelliptic locus is an immersion in characteristic not $2$. In characteristic $2$, we show the Torelli map restricted to the hyperelliptic locus fails to be an immersion because it is generically inseparable; moreover, the induced map on tangent spaces has kernel of dimension $g-2$ at every point.