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 $\Sym^{d} (X)$ 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.

2010 Mathematics Subject Classification: 14H40, 14J50

Keywords and Phrases: Symmetric product; automorphism; Torelli theorem.

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