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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,


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The particular family you mention is well discussed in the literature. My own recipe for looking at the familly locally is by writing the differential equation for its periods (see, e.g., D.R. Morrison's Picard--Fuchs equations and mirror maps for hypersurfaces and 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
The Hessian (the matrix of second derivatives) is invertible so the result follows from the (complex version of) the Morse lemma. – Torsten Ekedahl May 29 '10 at 14:00
For trackback purposes, the arxiv link to Morrison's paper is here – j.c. May 29 '10 at 16:57

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.

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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.

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That the lowest term of the Taylor expansion determines the singularity is not always true. It is true because the Hessian is non-singular in which case there is only one type as we are working over the complex numbers (it is over the reals that we have different indices). – Torsten Ekedahl May 30 '10 at 18:04
Indeed. I added this qualification, thanks for the comment. – Laie May 30 '10 at 19:42

and expanding the defining polynomial to lowest order gives you the equation describing the local neighborhood of the singular point.

Is there any reference for that?

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The proof in for instance Milnor: Morse theory of the real case works without changes in the complex case. – Torsten Ekedahl Jun 1 '10 at 4:09

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