I'm trying to understand the fundamental group of the following complete intersection in $RP^2 \times RP^2 \times RP^1$: \begin{eqnarray} &&t_1 \left( x_1^3 + x_2^3 + x_3^3 + a x_1 x_2 x_3 \right) + c t_2 (x_1 x_2 x_3) = 0 \\ &&t_2 \left( y_1^3 + y_2^3 + y_3^3 + b y_1 y_2 y_3 \right) + c t_1 (y_1 y_2 y_3) = 0 \end{eqnarray} where $x_1,x_2,x_3$ are the coordinates of one $RP^2$, $y_1,y_2,y_3$ are the coordinates of the other $RP^2$ and $t_1,t_2$ are the coordinates of $RP^1$. Also $a,b,c$ are real parameters.

I would also be interested in the fundamental group for special choices of the parameters, in case it is easier.

So far I have not been able to find any reference which helps me tackle this problem, so this would also be appreciated.

  • 1
    $\begingroup$ Why do you write hypersurface? It looks like your two equations are independent, and the locus they describe has codimension 2. $\endgroup$ – calc Dec 5 '13 at 21:33

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