Automorphism group of an algebraic curve (reviewed)

An automorphism of an algebraic curve is an isomorphism from the curve to itself. The set of automorphisms of a curve $X$ form a group $\mathrm{Aut}(X)$ under composition; this is the automorphism group of the curve.

The automorphism group of a genus 2 curve necessarily includes the hyperelliptic involution $(x,y)\mapsto(x,-y)$, which is an automorphism of order 2; this means that the automorphism group of a genus 2 curve is never trivial.

The geometric automorphism group of a curve $X/k$ is the automorphism group of $X_{\bar k}$.