Automorphisms of a symmetric product of a curve

TL;DR

This paper proves that for a smooth projective curve of genus greater than 2, all automorphisms of its symmetric product are derived from automorphisms of the curve itself, under certain conditions.

Contribution

It establishes that automorphisms of symmetric products of curves are completely determined by automorphisms of the original curve for sufficiently large d.

Findings

All automorphisms of Sym^d(X) are induced by automorphisms of X for d > 2g-2.

Automorphisms of the symmetric product mirror those of the original curve.

The result holds for curves of genus greater than 2.

Abstract

We show that all the automorphisms of the symmetric product Sym^d(X), d>2g-2, of a smooth projective curve X of genus g>2 are induced by automorphisms of X.