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

Abstract

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.