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

### Indranil Biswas

### Tomás L. Gómez

## 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.

## Cite this article

Indranil Biswas, Tomás L. Gómez, Automorphisms of a Symmetric Product of a Curve (with an Appendix by Najmuddin Fakhruddin). Doc. Math. 22 (2017), pp. 1181–1192

DOI 10.4171/DM/591