Theta-duality on Prym varieties and a Torelli theorem (Q2847187)

Let \(\widetilde C \to C\) be an étale double cover of a smooth projective curve of genus \(g\), and let \(P\) be the corresponding Prym variety. There are natural closed subschemes \(V^r \subset P\), which are Prym versions of the classical Brill--Noether loci. As a set, \(V^r\) consists of classes \(L \in \mathrm{Pic}^{2g-2}(\widetilde C)\) whose image under the norm map is the canonical class on \(C\), such that \(h^0(\widetilde C, L) \geq r+1\), and \(h^0(\widetilde C, L) \equiv r+1 \pmod 2\); by translating by some fixed element of \(V^{-1}\), these can be considered as subsets of \(P\). For example, \(V^1\) defines (after translation) a theta divisor on \(P\), and \(V^3\) is given by the stable singularities of the theta divisor. Let \(X^+\) (resp. \(X^-\)) be the inverse image of \(V^{1}\) (resp. \(V^0\)) in the symmetric power \(\widetilde C^{(2g-2)}\). The varieties \(X^+\) and \(X^-\) with their maps to \(P\) are Prym-analogues of the varieties \(C^{(g-1)}\) and \(C^{(g)}\) with their maps to the Jacobian of \(C\).NEWLINENEWLINEOn a ppav \((A,\Theta)\) one can define the \textit{theta-dual} of a subscheme \(X \subset A\), which is again a subscheme of \(A\) and can be described set-theoretically as \(T(X) = \{a \in A \, : \, a + X \subset \Theta\}\). It is known that for the classical Brill--Noether loci \(W_r\) in a Jacobian, one has the duality \(T(W_{d}) = W_{g-d-1}\). The behavior of the Brill--Noether loci in a Prym variety under theta-duality was studied by Welters, who proved that for a generic Prym of high enough dimension, \(T(V^3)\) has a unique \(2\)-dimensional component which is given by \(\widetilde C - \widetilde C\). Debarre proved that \(T(V^3)\) coincides with this surface when \(C\) is a general tetragonal curve of large enough genus. (See the paper for references.)NEWLINENEWLINEThe first main theorem of this paper is the equality \(T(V^2) = \widetilde C\), \(T(\widetilde C) = V^2\), if \(C\) is \textit{any} non-hyperelliptic curve of genus \(g \geq 4\).NEWLINENEWLINEThe second main theorem is the ``Torelli theorem'' of the title: it asserts that the variety \(X^-\) determines the pair \((\widetilde C, C)\). The analogous statement for \(X^+\) was known previously by \textit{J. C. Naranjo} [J. Reine Angew. Math. 560, 221--230 (2003; Zbl 1047.14014)] and \textit{R. Smith} and \textit{R. Varley} [Int. J. Math. 13, No. 1, 67--91 (2002; Zbl 1068.14030)].