Fundamental group of the smooth locus of a normal algebraic surface is a quotient of that of a Zariski open subset

Question

Let $X$ be a normal algebraic surface (over $\mathbb{C}$) and $Y$ its smooth locus, i.e., the complement of the singularities of $X$. Suppose $Z\subset Y$ is a Zariski open subset of $X$. Then is it true that $\pi_1(Y)$ is a quotient of $\pi_1(Z)$? (For computing the fundamental group, I'm considering not the Zariski topology but the usual topology.)

I've read a speical case of the above situation in https://projecteuclid.org/journals/nihonkai-mathematical-journal/volume-29/issue-2/Log-del-Pezzo-surfaces-of-rank-one-containing-the-affine/nihmj/1565048776.full, above Corollary 1.1: "If a normal algebraic surface contains the affine plane $\mathbb{C}^2$ as a Zariski open subset, then its smooth locus is simply-connected." I've asked for a proof of this statement in https://math.stackexchange.com/questions/4970345/complex-surface-containing-bbb-c2-as-a-zariski-open-subset, but I didn't get an answer.

Also, I've seen that for a normal algebraic surface containing $\mathbb{C}\times \mathbb{C}^*$ as a Zariski open subset, where $\mathbb{C}^*=\mathbb{C}-\{\text{point}\}$, the fundamental group of its smooth locus is a quotient group of $\pi_1(\mathbb{C}\times \mathbb{C}^*)=\mathbb{Z}$ (but I cannot remember where I've read this).

So it seems that the statement in the first paragraph is true, but I cannot see why. Can this be deduced from a van Kampen theorem argument?

Answer 1

The question is essentially asking why the inclusion of $Z$ into the smooth locus $Y$ induces a surjection on $\pi_1$. The ambient space $X$ is not really relevant here. You're removing some Zariski closed set $C = Y-Z$ from the complex manifold $Z$, and asking whether every loop in $Z$ is homotopic to one in $Y$ (relative to a basepoint). Since $C$ is Zariski closed, it's a union of lower-dimensional complex manifolds (namely its smooth locus, and the smooth locus of its singular set, etc.). It now follows by transversality that you can, indeed, homotope each loop in the desired manner. (Note here that complex is important; if one of these manifolds had codimension 1, then transversality would only guarantee a homotopic loop that intersects the codimension 1 submanifold in a point.)

I'm not sure what the best reference for this is. In my paper with Florentino and Lawton (https://arxiv.org/abs/1412.0272, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 17 (2017), no. 1, 143-185), we needed this fact, and included a brief discussion, pointing back to an earlier paper of mine. There are probably other places where this is explained.