Galois subspaces for smooth projective curves

Abstract

Given an embedding of a smooth projective curve $X$ of genus $g\ge 1$ into ${\mathbb{P}}^N$, we study the locus of linear subspaces of ${\mathbb{P}}^N$ of codimension 2 such that projection from said subspace, composed with the embedding, gives a Galois morphism $X\to {\mathbb{P}}^1$. For genus $g\ge 2$ we prove that this locus is a smooth projective variety with components isomorphic to projective spaces. If $g=1$ and the embedding is given by a complete linear system, we prove that this locus is also a smooth projective variety whose positive-dimensional components are isomorphic to projective bundles over étale quotients of the elliptic curve, and we describe these components explicitly.

To appear in Journal of Algebra. 17 pages including appendix. Comments are welcome!