Given a family of quintic hypersurfaces in $\mathbb{CP}^4$ by $x_1^5+x_2^5+x_3^5+x_4^5+x_5^5+(5+\epsilon)x_1x_2x_3x_4x_5$ we get a singular variety for $\epsilon=0$ with 125 singular points. I know from literature that the neighborhood of each singularity looks locally like the variety given by the equation $z_1^2+z_2^2+z_3^2+z_4^2=0$, which is a cone over $S^2\times S^3$. But how do I see that the neighbourhood is given by the above equation?

Thanks in advance,

Peter

Picard--Fuchs equations and mirror maps for hypersurfacesand his other papers placed in the arXiv) and then comparing the resulted DE in the neighbourhood of your singular point with the one for the cone. – Wadim Zudilin May 29 '10 at 5:31