Hilbert schemes of abelian group orbits.

*(English)*Zbl 1104.14003Let \(k\) be an arbitrary algebraically closed field. For any finite subgroup \(G\) of \(\text{SL}(3,k)\) of order prime to the characteristic of the ground field \(k\) the quotient space \(\mathbb{A}^3_k/G\) is then a normal Gorenstein variety with trivial canonical sheaf. In birational geometry, a natural choice of a crepant smooth resolution of the singular variety \(\mathbb{A}^3_k/G\) would be very convenient, but it seems that the general theories in this framework, such as the theory of minimal models and the theory of torus embeddings, do not canonically provide a resolution of this type.

In the paper under review, the author approaches this problem by studying a particular Hilbert scheme \(\text{Hilb}^G(\mathbb{A}^3_k)\), which finally turns out to be a canonical crepant resolution of the quotient variety \(\mathbb{A}^3_k/G\). The so-called \(G\)-orbit Hilbert scheme \(\text{Hilb}^G:= \text{Hilb}^G(\mathbb{A}^3_k)\) is, by definition, the scheme parametrizing all \(G\)-invariant smoothable \(0\)-dimensional subschemes of \(\mathbb{A}^3_k\) of length \(n:=|G|\). This object, which may be regarded as a certain substitute for the quotient \(\mathbb{A}^3_k/G\) was introduced by the author and Y. Itô in [Proc. Japan Acad., Ser. A 72, No. 7, 135–138 (1996; Zbl 0881.14002)] as a suitable tool in the study of resolutions of quotient singularities via the famous McKay correspondence.

In the present paper, the \(G\)-orbit Hilbert scheme \(\text{Hilb}^G\) is described for a finite abelian subgroup \(G\) of \(\text{SL}(3,k)\) resulting in the fact that \(\text{Hilb}^G\) appears then as a smooth torus embedding associated to a crepant fan in \(\mathbb{R}^3\) with apices constructed from the group \(G\). Furthermore, it is shown that the commutativity of \(G\) implies the nonsingularity of that associated fan.

This finally establishes the author’s main theorem (Theorem 0.1.) stating the following: For any abelian subgroup \(G\) of \(\text{SL}(3,k)\) of order prime to the characteristic of the ground field \(k\) the \(G\)-orbit Hilbert scheme \(\text{Hilb}^G(\mathbb{A}^3_k)\) is a crepant resolution of the quotient space \(\mathbb{A}^3_k/G\).

The first half of the present article is devoted to describing a \(G\)-orbit Hilbert scheme as a toric variety in arbitrary dimension. This part is based on a fine analysis of the corresponding lattices and \(G\)-graphes, which is highly interesting and important for its own sake. In the second half of the paper, the particular case of dimension three and an abelian subgroup \(G\) of \(\text{SL}(3,k)\) is inspected more closely by means of the special appearing \(G\)-graphs, culminating in the author’s main theorem mentioned above. At the end of the article, the author discusses some interesting examples in dimension three and four, thereby illustrating the variety of possibilities that can occur already in those low-dimensional cases.

In a sense, the present work may be regarded as a complement to the related earlier results by Y. Itô and M. Reid [in: Higher-dimensional complex varieties. Proc. Int. Conf. Trento, Italy, June 15–24, 1994. 221–240 (1996; Zbl 0894.14024)] and by Y. Itô and H. Nakajima [Topology 39, 1155–1191 (2000; Zbl 0995.14001)].

In the paper under review, the author approaches this problem by studying a particular Hilbert scheme \(\text{Hilb}^G(\mathbb{A}^3_k)\), which finally turns out to be a canonical crepant resolution of the quotient variety \(\mathbb{A}^3_k/G\). The so-called \(G\)-orbit Hilbert scheme \(\text{Hilb}^G:= \text{Hilb}^G(\mathbb{A}^3_k)\) is, by definition, the scheme parametrizing all \(G\)-invariant smoothable \(0\)-dimensional subschemes of \(\mathbb{A}^3_k\) of length \(n:=|G|\). This object, which may be regarded as a certain substitute for the quotient \(\mathbb{A}^3_k/G\) was introduced by the author and Y. Itô in [Proc. Japan Acad., Ser. A 72, No. 7, 135–138 (1996; Zbl 0881.14002)] as a suitable tool in the study of resolutions of quotient singularities via the famous McKay correspondence.

In the present paper, the \(G\)-orbit Hilbert scheme \(\text{Hilb}^G\) is described for a finite abelian subgroup \(G\) of \(\text{SL}(3,k)\) resulting in the fact that \(\text{Hilb}^G\) appears then as a smooth torus embedding associated to a crepant fan in \(\mathbb{R}^3\) with apices constructed from the group \(G\). Furthermore, it is shown that the commutativity of \(G\) implies the nonsingularity of that associated fan.

This finally establishes the author’s main theorem (Theorem 0.1.) stating the following: For any abelian subgroup \(G\) of \(\text{SL}(3,k)\) of order prime to the characteristic of the ground field \(k\) the \(G\)-orbit Hilbert scheme \(\text{Hilb}^G(\mathbb{A}^3_k)\) is a crepant resolution of the quotient space \(\mathbb{A}^3_k/G\).

The first half of the present article is devoted to describing a \(G\)-orbit Hilbert scheme as a toric variety in arbitrary dimension. This part is based on a fine analysis of the corresponding lattices and \(G\)-graphes, which is highly interesting and important for its own sake. In the second half of the paper, the particular case of dimension three and an abelian subgroup \(G\) of \(\text{SL}(3,k)\) is inspected more closely by means of the special appearing \(G\)-graphs, culminating in the author’s main theorem mentioned above. At the end of the article, the author discusses some interesting examples in dimension three and four, thereby illustrating the variety of possibilities that can occur already in those low-dimensional cases.

In a sense, the present work may be regarded as a complement to the related earlier results by Y. Itô and M. Reid [in: Higher-dimensional complex varieties. Proc. Int. Conf. Trento, Italy, June 15–24, 1994. 221–240 (1996; Zbl 0894.14024)] and by Y. Itô and H. Nakajima [Topology 39, 1155–1191 (2000; Zbl 0995.14001)].

Reviewer: Werner Kleinert (Berlin)