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In his paper "Smooth models for elliptic threefolds" (In: The Birational Geometry of Degenerations, Progress in Mathematics, v. 29, Birkhauser, (1983), 85-133), Rick Miranda mentions in the example of section 8 (page 101-102) that it is an unfortunate fact of life that there are no small resolutions for the singularity of the threefold defined by the equation

$$y^2=u^2+v^2+w^3\quad \text{in}\quad \mathbb{C}^4.$$

He said that it is a theorem of Brieskorn but he does not give a reference. Anyone has a reference to this theorem and its proof?

Updated (following some of the answers, the question can be generalized right away):

More generally the same question can be asked for

$$y^2=u^2+v^2+w^{2k+1}\quad \text{in}\quad \mathbb{C}^4, \quad\text{where}\\quad k\in \mathbb{N}_0.$$

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3 Answers 3

up vote 8 down vote accepted

A singularity of type $y^2=u^2+v^2+w^k$ admits a small resolution if and only if $k$ is even. If $k$ is odd the corresponding singularity is factorial, so there is no small resolution. See the paper of Friedman "Simultaneous resolution of 3-fold double points", p. 675.

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This answer to a more general question might also be relevant for similar questions.

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It is worth noting the following result by S. Katz:

If the singularity $X$ defined by $xy-g(z,t)$ is isolated (so $g=0$ is reduced) and $cA_n$ (meaning a general hyperplane section is a type $A_n$ singularity) then $X$ has a small resolution iff the curve $g=0$ has $n+1$ distinct branches.

The statement you want is the case $n=1$: $z^2+t^k=0$ has $2$ branches iff $k$ even.

I learned about this result in this paper, where the authors proved the non-commutative(!) version of Katz's theorem (see Theorem 5.5).

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Thanks a lot for the Katz's reference! –  JME Jul 26 '10 at 12:55
    
@JME: You're welcome! –  Hailong Dao Jul 26 '10 at 14:27

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