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.
Best Answer
I would like to mention one more homotopy invariant of smooth projective varieties that is also a birational invariant. If $X$ is a smooth projective variety, then the torsion subgroup $T(X)$ of ${\rm H}^3(X,\mathbb{Z})$ is a birational invariant. This is explained in the beautiful paper "Some elementary examples of unirational varieties which are not rational" by Artin-Mumford, which starts exactly outlining a "homotopical" approach to showing that there exist unirational threefolds that are not rational. Their homotopic criterion is that rational varieties $X$ have trivial group $T(X)$ and they construct unirational varieties with non-trivial two-torsion in such group, showing that these are examples of unirational, non rational varieties.
Finally, for surfaces the quantity $K^2+\rho$ is a birational invariant, where $K^2$ is the self intersection of the canonical divisor class on $X$ and $\rho$ is the rank of the N\'eron-Severi group of $X$). This is not too deep, as it is immediate knowing that a birational map of surfaces is a composition of blow ups and blow downs of smooth points (which for surfaces is quite easy!) and the above quantity is obviously invariant under a blow up. The reason for mentioning it, is that the N\'eron-Severi group can be defined in terms of Hodge theory, which, while not entirely homotopical, certainly has a homotopical feel to it. Moreover, many birational invariants are defined using Hodge structures, so it seemed useful to point it out!