8
$\begingroup$

I have an explicit question.

I have a complex projective variety defined by $2\times 2$ minors of a matrix. The entries are polynomials from a weighted projective space. In fact, it's a singular 3-fold with only quotient singularities.

Now I want to show that it is simply connected.

Can anyone help in giving direction in this regard?

$\endgroup$
2
  • 3
    $\begingroup$ you could have a look at arxiv.org/abs/1311.6086 On the fundamental group of a variety with quotient singularities Indranil Biswas, Amit Hogadi $\endgroup$
    – Niels
    Commented Jul 5, 2014 at 7:29
  • 3
    $\begingroup$ Any variety is defined by 2x2 minors in a 'sufficiently ample' embedding. This is due to Mumford and he shows that under a sufficiently high Veronese re-embedding the variety is the intersection of a linear subspace and the equations of the Veronese which are quadratic, and in fact, are determinantal. $\endgroup$
    – meh
    Commented Jul 7, 2014 at 2:41

1 Answer 1

3
$\begingroup$

I am not sure this will help, but the methods we use in Fundamental Group of Moduli Spaces of Representations might be useful (if you are working over $\mathbb{C}$).

In particular, these two properties are worth pointing out:

  1. Given any non-empty Zariski open subset $U \subset V$, where $V$ is normal, the homomorphism $\pi_1(U) \to \pi_1(V)$ induced by the inclusion map is surjective.
  2. The GIT quotient map $V\to V//G$ is $\pi_1$-surjective, where $G$ is a connected reductive affine algebraic group.

Another tool that might be of interest is Proposition 5.8 which gives three conditions for the inclusion map $V\setminus W\hookrightarrow V$ to be 2-connected (where $W\subset V$ is a subvariety).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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