The Torelli map restricted to the hyperelliptic locus

The classical Torelli theorem states that the Torelli map, ending a curve to its Jacobian, is injective on points. However, the Torelli map is not injective on tangent spaces at points corresponding to hyperelliptic curves. This leads to the natural question: If one restricts the Torelli map to the locus of
hyperelliptic curves, is it then an immersion?

We give a complete answer to this question, starting out by describing the classical history and several surprising foundational gaps in the literature. Along the way, we will learn about Shinichi Mochizuki's valuative criterion for locally closed immersions and its relation to Brian Conrad's
library app idea.