Quotient curve (reviewed)

Let $X$ be an algebraic curve and let $H$ be a finite subgroup of its automorphism group.

The quotient curve $X/H$ is the algebraic curve obtained by identifying points of $X$ that lie in the same $H$-orbit (equations defining $X/H$ as an algebraic variety of dimension one can be constructed from the equations defining $X$ and the automorphisms in $H$).

The natural projection $X\to X/H$ that sends each point on $X$ to its $H$-orbit is a surjective morphism