Resolution of singularities of affine normal quasihomogeneous \(SL_ 2\)- varieties.

*(English. Russian original)*Zbl 0703.14031
Funct. Anal. Appl. 22, No. 4, 338-339 (1988); translation from Funkts. Anal. Prilozh. 22, No. 4, 94-95 (1988).

An algebraic variety with the action of an algebraic group G as group of algebraic morphisms is called quasihomogeneous if it has an open orbit. The author considers the case of \(G=SL_ 2\). The 2-dimensional case is well known. The remaining 3-dimensional case was treated by H. Kraft [“Geometrische Methoden in der Invariantentheorie” (1985; Zbl 0669.14003)] when the stabilizers of all points are trivial. Here the author removes this restriction on stabilizers. 3-dimensional quasihomogeneous \(SL_ 2\)-varieties are classified by two numbers: the order m of the stabilizer of a point of the open orbit and a rational number h \((0<h<1)\) called the height [V. L. Popov, Izv. Akad. Nauk SSSR, Ser. Mat. 37, 1038-1055 (1973; Zbl 0251.14018)].

The author gives an explicit description of the variety \(E_{h,m}\) with given order and height as the closure of the orbit of a vector in certain representation of \(SL_ 2\). The main theorem asserts the existence of a minimal equivariant resolution with the following structure of the exceptional locus: it is either \({\mathbb{P}}^ 1\) or a union of ruled surfaces intersecting along fibres on the rulings. An algorithm is described in order to find discrete data determining these surfaces. The author indicates that the resolution is obtained as \(SL_{2*B}\tilde Y\) where B is the Borel subgroup and \(\tilde Y\) is a B-equivariant resolution of a toric variety constructed from \(E_{h,m}\) (which is the closure of the orbit of the Borel subgroup of some correctly chosen vector).

The author gives an explicit description of the variety \(E_{h,m}\) with given order and height as the closure of the orbit of a vector in certain representation of \(SL_ 2\). The main theorem asserts the existence of a minimal equivariant resolution with the following structure of the exceptional locus: it is either \({\mathbb{P}}^ 1\) or a union of ruled surfaces intersecting along fibres on the rulings. An algorithm is described in order to find discrete data determining these surfaces. The author indicates that the resolution is obtained as \(SL_{2*B}\tilde Y\) where B is the Borel subgroup and \(\tilde Y\) is a B-equivariant resolution of a toric variety constructed from \(E_{h,m}\) (which is the closure of the orbit of the Borel subgroup of some correctly chosen vector).

Reviewer: A.Libgober

##### MSC:

14L30 | Group actions on varieties or schemes (quotients) |

14E15 | Global theory and resolution of singularities (algebro-geometric aspects) |

14M17 | Homogeneous spaces and generalizations |

##### Keywords:

3-dimensional quasihomogeneous \(SL_ 2\)-varieties; stabilizer of a point of the open orbit; height; minimal equivariant resolution
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\textit{D. I. Panyushev}, Funct. Anal. Appl. 22, No. 4, 338--339 (1988; Zbl 0703.14031); translation from Funkts. Anal. Prilozh. 22, No. 4, 94--95 (1988)

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##### References:

[1] | V. L. Popov, ”Quasihomogeneous affine algebraic varieties of the group SL2,” Izv. Akad. Nauk SSSR, Ser. Mat.,37, No. 5, 792-832 (1973). · Zbl 0281.14022 |

[2] | H. Kraft, Geometric Methods in the Theory of Invariants [Russian translation], Mir, Moscow (1987). · Zbl 0669.14004 |

[3] | D. Luna and Th. Vust, ”Plongements d’espaces homogĂ¨nes,” Comment. Math. Helvetici,58, No. 2, 186-245 (1983). · Zbl 0545.14010 · doi:10.1007/BF02564633 |

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