Spectral curves have acquired a central role in understanding the moduli spaces of vector bundles and Higgs bundles on a curve. A Higgs G-bundle on an arbitrary variety S (together with some additional data, such as a representation of G) determines a spectral cover S of S and an equivariant sheaf on S. The purpose of these notes is to combine and review various results about spectral covers, focusing on the decomposition of their Picards (and the resulting Prym identities) and the interpretation of a distinguished Prym component as parameter space for Higgs bundles.

1. Introduction Spectral curves arose historically out of the study of differential equations of Lax type. Following Hitchin's work [HI], they have acquired a central role in understanding the moduli spaces of vector bundles and Higgs bundles on a curve. Simpson's work [S] suggests a similar role for spectral covers S of higherdimensional varieties S in moduli questions for bundles on S.

The purpose of these notes is to combine and review various results about spectral covers, focusing on the decomposition of their Picards (and the resulting Prym identities) and the interpretation of a distinguished Prym component as parameter space for Higgs bundles. Much of this is modeled on Hitchin's system, which we recall in Section 1, and on several other systems based on moduli of Higgs bundles, or vector bundles with twisted endomorphisms, on curves. By peeling off several layers of data that are not essential for our purpose, we arrive at the notions of an abstract principal Higgs bundle and a cameral (roughly, a principal spectral) cover.