Automorphisms of a Symmetric Product of a Curve (with an Appendix by Najmuddin Fakhruddin)

Let X be an irreducible smooth projective curve of genus g>2 defined over an algebraically closed field of characteristic different from two. We prove that the natural homomorphism from the automorphisms of X to the automorphisms of the symmetric product \mathrm{Sym}^d(X) is an isomorphism if d>2g-2 . In an appendix, Fakhruddin proves that the isomorphism class of the symmetric product of a curve determines the isomorphism class of the curve.