Deformation with fixed ramification

Question

Suppose that $f : X \to Y$ is a finite, surjective morphism of normal varieties. I want to know about the space of first-order deformations of $X$ over $Y$ with fixed ramification, i.e. the deformations of $f$ with fixed target $Y$, such that the (set-theoretic) image of the ramification divisor $f(R_t)$ is constant.

I think that when $X$ and $Y$ are smooth, there are no such deformations: the normal sheaf $N_f$ is supported on the ramification divisor $R \subset X$ (Sernesi, "Deformations...", pg 171), and a deformation of $f$ forces the ramification to move too. But I am not sure if I can dispense with the smoothness assumption (or indeed whether my question is well-posed without it?)

Answer 1

The moduli problem you are interested in is zero-dimensional in characteristic zero. (In this answer we will work over $\mathbb{C}$.)

Indeed, for all $d\geq 1$, and all normal varieties $X$ and $Y$, the set of isomorphism classes of finite degree $d$ surjective morphisms $X\to Y$ ramified over a fixed closed subset $B\subset Y$ is finite. This follows from the fact that $\pi_1(Y^{an})$ is finitely generated. (The fact that $d$ is fixed is because you are deforming a fixed finite map $X\to Y$. Thus, $d$ equals $\deg(X\to Y)$.)

The fact that $\pi_1(S^{an})$ is finitely generated holds for any variety $S$ over $\mathbb{C}$; see SGA7.I Théorème 2.3.1 Expose II.

In case you are interested: there is a (small) difference in studying finite etale covers of a variety $U = Y\setminus D$ and studying finite surjective maps $X\to Y$ ramified only over $D$. Indeed, there is a fully faithful functor from the category of finite etale covers of $U$ to the category of finite surjective morphism $X\to Y$ ramified only over $D$. (To $V\to U$ one associates the normalization of $Y$ in the "function field" of $V$.) This functor is not essentially surjective (because you can sometimes extend a given $V\to U$ to a finite surjective map $X\to Y$ with $Y$ normal but $X$ non-normal. Think of a rational function on a nodel curve.) However, if you stick to normal varieties, the category you are interested in is indeed equivalent to the category of finite etale covers of $Y\setminus B$.