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The universal difference variety over \(\overline{\mathcal M}_g\). (English) Zbl 1274.14031

The authors compute the dimension of the Deligne-Mumford compactification of the universal difference variety. For a curve of genus \(g\), if \(i=\lfloor \frac{g+1}{2}\rfloor\) then we always have a surjective map \(C_i\times C_i\to J(C)\). In the case that \(g\) is even, the map is finite and if \(i\) is odd, then \(C_{i-1}\times C_{i-1}\to J(C)\) gives a natural divisor on \(J(C)\). These maps can be interpreted, on a non-hyperelliptic curve, as a resolution of singularities for a natural divisor on \(J(C)\), defined by setting \(Q_C\) to be the dual of the kernel of \(H^0(C,K_C)\otimes \mathcal{O}_C\to K_C\), and taking the divisor to be \(\Theta_{\bigwedge^{i-1}Q_C}=\{\xi\in J(C)|h^0(C,\bigwedge^{i-1}Q_C\otimes \xi)\geq 1\}\). This resolution is analogous to the resolution of the classical theta divisor by the Abel-Jacobi map \(C_{g-1}\to \mathrm{Pic}^{g-1}(C)\).
The authors are motivated by the connection to Green’s conjecture, which on a curve of odd genus and maximal Clifford index is true if and only if the natural map \(\bigwedge^{i-1}H^0(C,K_C)^\vee\to H^0(C,\bigwedge^{i-1}Q_C)\) is an isomorphism, and the codomain is related to the singularities of \(\Theta_{\bigwedge^{i-1}Q_C}\) by [Y. Laszlo, Duke Math. J. 64, No. 2, 333–347 (1991; Zbl 0753.14023)].
Additionally, they construct a divisor on \(\overline{\mathcal{M}}_{g,g-3}\), compute its class, and use it to find a class on \(C_{g-3}\) which then they conjecture to span an extremal ray in the effective cone of \(C_{g-3}\).

MSC:

14H10 Families, moduli of curves (algebraic)
14H40 Jacobians, Prym varieties
14H42 Theta functions and curves; Schottky problem

Citations:

Zbl 0753.14023
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References:

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