The Torelli map restricted to the hyperelliptic locus

The classical Torelli theorem states that the Torelli map, sending a curve to its Jacobian, is an injection 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 will give a complete answer to this question, starting out by describing the classical history.