TITLE: Birational invariants and fundamental groups
QUESTION [28 upvotes]: In pondering this MO question and people's efforts to answer it, and recalling also something that I learned in my youth about using Morse theory ideas to prove some results of Lefschetz in the complex case, I seem to have learned two things -- things that I suppose are absorbed in the cradle by those who study algebraic geometry as opposed to learning it by osmosis -- or maybe I haven't got them quite right. Anyway:
(1) Blowing up a surface at a smooth point does not change the fundamental group, and (therefore ?) the fundamental group of a smooth projective surface is a birational invariant.
(2) Surfaces in $\mathbb P^3$ are simply connected, and more generally for a set $X\subset\mathbb P^n$ defined by a single homogeneous equation of degree $>0$ the pair $(\mathbb P^n,X)$ is at least $(n-1)$-connected. That is, the relative homotopy groups and therefore the relative homology groups vanish up through dimension $n-1$.)
(This is all over the complex numbers.) 
Is this correct? And, taking off from (1), what are some other simple statements about invariants from homotopy theory that are birational invariants? And what are the first things to know about birational invariants that do not come from topology?
EDIT  I wish I could accept more than one answer.

REPLY [6 votes]: The paper of Keum and Zhang 
"Fundamental groups of open K3 surfaces, Enriques surfaces and Fano 3-folds", 
Journal of Pure and Applied Algebra vol. 170
provides some answers to Tom's question "What happens to the fundamental group of a singular variety when removing the singular points?"
It appears that the fundamental group of the smooth part can be quite complicate also in simple situations. For instance, one of the results in the paper is the following:
"THEOREM. Let X be a $K3$ surface with at worst Du Val singularities (then X is still simply connected), and let $X^0$ be its smooth part. The number $c$=#(Sing $X$) is bounded by $16$, and if $c=16$ then $\pi_1(X^0)$ is infinite".
So, given for instance a quartic surface $X \subset \mathbb{P}^3$ with $16$ nodes (a Kummer surface) we have 
$\pi_1(X)=\{1\}$, but $\pi_1(X^0)$ is infinite! 
This follows from the fact that $X^0$ has an étale $\mathbb{Z}_2$-cover $Y^0 \to X^0$, where $Y^0$ is an Abelian surface minus 16 points.
For smaller values of $c$, the group $\pi_1(X^0)$ is finite, but not trivial in general. 
In higher dimension, there is the following
"CONJECTURE. Let $V$ be a $\mathbb{Q}$-Fano $n$-fold. Then the topological fundamental group $\pi_1(V^0)$ of the smooth part $V^0$ of $V$ is finite". 
(a normal variety $V$ with at worst log terminal singularities is $\mathbb{Q}$-Fano if, by definition, the anti-canonical divisor $−K_V$ is $\mathbb{Q}$-Cartier and ample).