Given an embedding of a smooth projective curve $X$ of genus $g\geq1$ 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 $g\geq2$ 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 etale quotients of the elliptic curve, and we describe these components explicitly.

中文翻译：

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平滑投影曲线的伽罗瓦子空间

给定类 $g\geq1$ 的平滑投影曲线 $X$ 嵌入 $\mathbb{P}^N$，我们研究了辅维 2 的 $\mathbb{P}^N$ 的线性子空间的轨迹，例如来自所述子空间的投影，由嵌入组成，给出了伽罗瓦态射 $X\to\mathbb{P}^1$。对于属 $g\geq2$，我们证明该轨迹是一个光滑的射影簇，其分量与射影空间同构。如果 $g=1$ 并且嵌入由一个完整的线性系统给出，我们证明这个轨迹也是一个光滑的射影簇，它的正维分量同构于椭圆曲线的 etale 商上的射影丛，我们描述了这些组件显式。