Torelli Theorem for moduli stacks of vector bundles on tame Deligne Mumford curves

Abstract

We prove a Torelli theorem for full moduli stacks of vector bundles of rank at least three with fixed determinant on a smooth, proper, geometrically connected, tame Deligne--Mumford curve of coarse genus at least two over an arbitrary field $k$. We also prove a Torelli theorem for the full moduli stack of vector bundles of rank at least three of fixed-degree under an explicit compatibility hypothesis. The method is intrinsic to the stack: faithful special-linear linearizations recover Picard stacks, and determinants of tangent complexes recover the canonical principal polarization. This is the first part in a series of two papers.

arXiv comment56 pages. v1 has been broken up into two separate papers to enhance readability. Comments welcome!