Extension groups of tautological bundles on symmetric products of curves

We provide a spectral sequence computing the extension groups of tautological bundles on symmetric products of curves. One main consequence is that, if E ≠ O X is simple, then the natural map Ext 1 ( E , E ) → Ext 1 ( 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.