Algebraic fundamental group of a variety I have a very explicit question. Consider a projective variety (a Fano 3-fold) in $\mathbb P^{10}$ defined by 3 quadrics and 32 cubic equations. I want to show that the algebraic fundamental group of the variety is {1}. Is there a way to show it?
 A: I am just collecting my comments above as an answer.  Over $\mathbb{C}$ (or any algebraically closed field of characteristic $0$), every log $\mathbb{Q}$-Fano variety is simply connected.  A log $\mathbb{Q}$-Fano variety is a pair $(X,D)$ of a normal projective variety $X$ and an effective $\mathbb{Q}$-Cartier divisor $D$ on $X$ such that $(X,D)$ has at worst Kawamata log terminal singularities and $-(K_X+D)$ is nef and big.  For such a pair $(X,D)$, for every desingularization $\nu:\widetilde{X}\to X$, $\widetilde{X}$ is rationally connected: this is a theorem of Qi Zhang.
MR2208131 (2006m:14021) 
Zhang, Qi(1-MO) 
Rational connectedness of log Q-Fano varieties. (English summary)  
J. Reine Angew. Math. 590 (2006), 131–142. 
14E30 (14J45) 
https://arxiv.org/abs/math/0408301
This was reproved by Hacon and McKernan 
as part of their proof of the Shokurov conjecture.
For a smooth projective variety that is rationally connected in characteristic $0$, or more generally if it is separably rationally connected in any characteristic, the variety is algebraically simply connected.  This was first proved over $\mathbb{C}$ by Campana (who also proves that the topological fundamental group of the underlying complex manifold is finite, so that the complex manifold is simply connected).  In positive characteristic this was proved by Kollár.  One nice reference is Section 3.4 of the following.
MR2074059 (2005g:14096) 
Debarre, Olivier(F-STRAS-I) 
Variétés rationnellement connexes (d'après T. Graber, J. Harris, J. Starr et A. J. de Jong). 
Séminaire Bourbaki. Vol. 2001/2002. 
Astérisque No. 290 (2003), Exp. No. 905, ix, 243–266.  
14M20 (14D06) 
https://eudml.org/doc/110309
Since normal varieties are unibranch, the fundamental group of $X$ is the image of the fundamental group of $\widetilde{X}$ (in general the surjective group homomorphism can have nontrivial kernel -- e.g., for cones over plane curves of degree $d\geq 3$).  Since $\widetilde{X}$ is simply connected, also $X$ is simply connected.
Of course it is also important to understand the fundamental group of the smooth locus of $X$.  I believe the best results for the smooth locus of $X$ are due to Chenyang Xu, who proves that the algebraic fundamental group of the smooth locus is finite.
