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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?

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    $\begingroup$ 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
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    $\begingroup$ 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

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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).

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