Extension groups of tautological bundles on symmetric products of curves

Abstract

We provide a spectral sequence computing the extension groups of tautological bundles on symmetric products of curves. One main consequence is that, if E≠ OX is simple, then the natural map Ext1(E,E)→Ext1(E[n],E[n]) is injective for every n. Along with previous results, this implies that E↦ E[n] defines an embedding of the moduli space of stable bundles of slope μ∉ [- 1 , n- 1] on the curve X into the moduli space of stable bundles on the symmetric product X(n). The image of this embedding is, in most cases, contained in the singular locus. For line bundles on a non-hyperelliptic curve, the embedding identifies the Brill–Noether loci of X with the loci in the moduli space of stable bundles on X(n) where the dimension of the tangent space jumps. We also prove that E[n] is simple if E is simple. © 2022, The Author(s).