Generic Torelli for \(W_ d^ r\) (Q2277035)

The classical Torelli theorem says that the polarized Jacobian of a curve completely determines the curve. By extension, one talks about Torelli problems as trying to identify a variety through some associated variety. A ``Torelli theorem'' says that the association map is injective while a ``generic Torelli theorem'' says that it is a birational morphism with its image. In this paper, we consider the association of a curve with the set of special divisors of fixed degree and dimension or \(W^ r_ d\). If the dimension of \(W^ r_ d\) is at least two, we show that generic Torelli holds. When the dimension of \(W^ r_ d\) is one, we can only show the result for \(r=1\). We expect it to be true for all values of r.