The link of a singular quintic hypersurface in CP^4 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
 A: Thank you very much.
I know that this family is well discussed, but I've never found a "proof" for the fact that this indeed is an $A_1$-singularity. 
The motivation is the following. I'm actually looking for singular CY-varieties that are not conifolds but have singularities with a neighborhood homeomorphic to a cone over the connected sum of $S^2\times S^3$.
This for example would be the case if the neighborhood of the singularity is given by $x_1^2+x_2^2+x_3^c+x_4^d=0$ in $\mathbb{C}^4$ for suitable c and d.
But I first will try to understand the paper on Picard-Fuchs equations.
A: The singularity type can be determined in a concrete and simple way by considering 
local coordinates $z_i$ around a singular point (with coordinates $x_i = a_i$, say). 
Writing these local coordinates as $x_i = a_i + z_i$ and expanding the defining 
polynomial to lowest order gives you the equation describing the local neighborhood 
of the singular point. The type of this singular point can then be determined from this 
local equation via the Hessian if it is nonsingular. 
