Fourier-Mukai transform for fine compactified Prym varieties

Abstract

Consider a finite covering ${\hat{I}}^2:C\to X$ of a smooth projective curve $X$ by a reduced, projective, planar curve $C$. Associated to two general polarizations on $C$, $q$ and $q^{\prime }$, one can construct the corresponding compactified Prym varieties ${\overline{\mathrm{P}}}_{\hat{I}}^2(q)$ and ${\overline{\mathrm{P}}}_{\hat{I}}^2(q^{\prime })$. Consider $\hat{I}“$ to be the group of line bundles whose torsion coincides with the order of ${\hat{I}}^2$. In this article we construct a Fourier-Mukai transform between the derived categories of ${\overline{\mathrm{P}}}_{\hat{I}}^2(q)$ and the $\hat{I}“$-equivariant derived category of ${\overline{\mathrm{P}}}_{\hat{I}}^2(q^{\prime })$. Hence, we obtain a derived equivalence between the $\mathrm{SL}(n,\mathbb{C})$-Hitchin fibre and its associated $\mathrm{PGL}(n,\mathbb{C})$-Hitchin fibre for a dense class of singular spectral curves. Our work then provides the extension of the Fourier-Mukai transform constructed by Arinkin and Melo-Rapagnetta-Viviani, which corresponds to autoduality of $\mathrm{GL}(n,\mathbb{C})$-Hitchin fibres in this class of singular spectral curves.