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Compound Du Val 3-fold singularities form a good class of singularities in 3-fold singularity theory. I would like to know which singularities admit crepant resolutions. If I remember correctly, $cA_{n}$ admits a crepant resolution. What about $cD_{n}$- and $cE_{n}$-singularities?

I would really appreciate it if you could give me a reference or explain what is known about crepant resolution of cDV singularities.

Thank you for your help.

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  • $\begingroup$ What's a compound du val singularity? Bridgeland-King-Reid have a paper treated 3-fold quotient singularities by finite subgroups of SL(3). I assume that's Du Val singularity for 3-folds? $\endgroup$
    – temp
    Commented Sep 19, 2012 at 5:19
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    $\begingroup$ It does not seem possible to say that $cA_n$ admits a crepant resolution. For example, $x^2+y^2+z^2+t^n=0$ is $cA_1$ for $n\ge 2$ but has a crepant resolution if and only if $n$ is even. $\endgroup$
    – inkspot
    Commented Sep 19, 2012 at 10:11
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    $\begingroup$ For an answer in the $cA_n$ case see Katz's paper: "Small resolutions of Gorenstein threefold" (small is equivalent to crepant in this case) . The $cD_n$ case seems a lot harder, it was discussed a bot there also, but I do not know any other references. $\endgroup$ Commented Sep 19, 2012 at 18:03
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    $\begingroup$ I think an isolated $cA_n$ singularity $xy + f(z,w)$ admits a crepant (= small) resolution if and only if $f(z,w)$ is a product of smooth curve germs. Existence is inductive blow up of ideals such as $(x,h(z,w))$ where $h$ is a factor of $f$. The converse requires some justifcation... $\endgroup$ Commented Apr 22, 2021 at 14:26
  • $\begingroup$ That's nice Evgeny! Please don't hesitate to write down whatever comes to your mind in an answer $\endgroup$
    – aglearner
    Commented Apr 22, 2021 at 15:50

1 Answer 1

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Background. The threefold compound du Val singularities have been introduced by Miles Reid in the 1980s [R1, R2, R3]. Their geometric description is that a general hyperplane section through the singular point is a du Val singularity. It has been proved by Miles Reid that isolated compound du Val singularities are precisely Gorenstein terminal singularities, and by definition of terminal, every crepant resolution is the same as small. Curiously, a Gorenstein threefold with a small resolution must have compound du Val singularities [R2, (1.2)]. On the other hand, compound du Val singularities always admit so-called $\mathbf{Q}$-factorializations [K, 4.5], which are more natural and less restrictive than small resolutions.

Compound $A_n$. I will consider the case of isolated $cA_n$ singularities, which are the simplest kind. They are characterized by the property that the completion at the singular point is isomorphic to $$k[[x,y,z,w]] / (xy + f(z,w))$$. The condition of being an isolated singularity is equivalent to $f(z,w)$ being reduced, and $n$ in the $cA_n$ stands for the lowest degree of monomial in $f$ minus one. Notation-wise, in the singularities and commutative algebra communities, the emphasis is on $A_n$ singularities $xy + z^2 + w^{n+1}$, which are among $cA_1$ in this definition.

Setting. Let $k$ be algebraically closed field of characteristic zero; in this case $xy$ after changing the basis is same as $x^2 + y^2$ (but $xy$ is the correct expression over non-closed fields too). We work with the local model $X = \mathrm{Spec}(k[x,y,z,w] / (xy + f(z,w))$. Existence of small resolutions is controlled by the existence of sufficiently many Weil divisors which are not Cartier (cf [K, Proof of 4.5]). Note that $\mathrm{Pic}(X) = 0$, because $k[X]$ is a normal graded ring. Thus to say that $X$ is factorial is the same as to say $\mathrm{Cl}(X) = 0$.

Claim 1. A factorial variety does not admit small resolutions, see this answer.

Claim 2. $\mathrm{Cl}(X) \simeq \mathbf{Z}^{r-1}$, where $r$ is the number of irreducible factors of $f(z,w)$. I think it should not be difficult to prove this directly, but I don't know the reference; for a proof using derived categories and Knorrer periodicity, see Lemma 2.22 in this paper. See below for the explicit Weil divisors on $X$.

Putting Claim 1 and Claim 2 together we obtain:

Corollary. If $f(z,w)$ is irreducible and non-smooth (e.g. $f(z,w) = z^2 + w^{2k+1}$), then $X$ admits no small resolutions (cf [R2, Cor. 1.16]).

Now the general result:

Proposition. $X$ admits a small resolution if and only if $f(z,w)$ is a product of distinct smooth curve germs.

Inductive construction for the "if" direction. Assume $f(z,w) = g(z,w) h(z,w)$, where $g(z,w)$ is a smooth curve germ, and $h(z,w)$ is coprime to $g(z,w)$. Consider the Weil divisor $D$ given by $x = g(z,w) = 0$. Let $\pi: \widetilde{X} \to X$ be the blow up of $D$. Writing the local charts of the blow up, we see that $\pi$ contracts a $\mathbf{P}^1$ to a singular point, and $\widetilde{X}$ has a singularity $xy + h(z,w) = 0$.

Obstruction to the "only if" direction. I think it follows from the fact that all $\mathbf{Q}$-factorializations are related by flops (Kawamata), and in particular existence of a small resolution implies that any $\mathbf{Q}$-factorialization is smooth. The inductive construction above will produce $\mathbf{Q}$-factorializations with singularities $xy + h(z,w) = 0$, with $h(z,w)$ irreducible nonsmooth. (One should come up with a more conceptual and direct proof which avoids using $\mathbf{Q}$-factorializations.)

Local vs global. One needs to be aware, that we have only constructed small resolutions in a local model, hence formally, or complex analytically. Indeed, many projective threefolds with $A_1$ singularities do not admit projective small resolutions, and this is also controlled by Weil divisors. For example $1$-nodal cubic hypersurfaces in $\mathbf{P}^4$ are factorial, hence by the Claim 1 do not admit projective small resolutions (cf [K, page 104]).

Classical references.

[R1] Miles Reid, "Canonical threefolds'', 1980

[R2] Miles Reid, "Minimal models of canonical threefolds'', 1983

[R3] Miles Reid, "Young person's guide to canonical singularities'', 1987

[K] Yujiro Kawamata, "Crepant Blowing-Up of 3-Dimensional Canonical Singularities and Its Application to Degenerations of Surfaces", 1988

Fun quote. I do not wish to go at present into the various interesting questions concerned with resolving the cDV points; for many purposes it seems natural to leave them alone! [R1] :-)

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  • $\begingroup$ Evgeny, thanks a lot for such a detailed answer! $\endgroup$
    – aglearner
    Commented Apr 28, 2021 at 23:55
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    $\begingroup$ I found that the main Proposition is the result of Katz from "Small resolutions of Gorenstein threefold Katz" (mentioned by Hailong Dao in the comments), who also does the $cD_n$ case. Katz interprets cDV singularties and their resolutions in terms of simultaneous resolutions of surface singularities and somehow reduces to the universal case $xy = (z-a_1 w) \cdots (z-a_{n+1} w)$. $\endgroup$ Commented Apr 30, 2021 at 14:18

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