Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $X^k$ be a complete intersection in $\mathbb CP^n$ of dimension $k>1$. Is it true that a smooth variety obtained by resolutions of singularities of $X$ is simply-connected?

Note that in the case $X^k$ is smooth itself, $\pi_1(X^k)=0$.

share|improve this question

1 Answer 1

up vote 5 down vote accepted

No. For example, take $X$ to be the projective cone in $\mathbb P^3$ of a smooth plane curve $C$ of degree at least $3$. Then blowing up the vertex gives a resolution that is a $\mathbb P^1$-bundle over $C$, and the fundamental group is that of $C$.

share|improve this answer
    
thanks a lot! Do you think there can be any positive statement in this direction? –  aglearner Nov 1 '12 at 0:47

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.