On automorphisms of algebraic curves

An irreducible, algebraic curve ${\mathcal{X}}_g$ of genus $g\ge 2$ defined over an algebraically closed field $k$ of characteristic $\mbox{char } , k = p \geq 0$,hasfiniteautomorphismgroup$\mbox{Aut} (\mathcal X_g)$. In this paper we describe methods of determining the list of groups $\mbox{Aut} (\mathcal X_g)$forafixed$g\geq 2$. Moreover, equations of the corresponding families of curves are given when possible.