Geodesic
The canonical module of a~quasihomogeneous normal affine $SL_2$-variety
Sbornik. Mathematics, Tome 73 (1992) no. 2, pp. 569-578

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For the varieties mentioned in the title a description is given for the canonical divisor, the Picard group, and the divisor class group. In particular, it follows from this that the singular 3-dimensional quasihomogeneous $SL_2$-varieties are not Gorenstein. The canonical module is described. All descriptions are given in terms of discrete parameters: the height and the degree of a quasihomogeneous affine $SL_2$-variety. 
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     author = {D. I. Panyushev},
     title = {The canonical module of a~quasihomogeneous normal affine $SL_2$-variety},
     journal = {Sbornik. Mathematics},
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     publisher = {mathdoc},
     volume = {73},
     number = {2},
     year = {1992},
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     url = {http://geodesic.mathdoc.fr/item/SM_1992_73_2_a16/}
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D. I. Panyushev. The canonical module of a~quasihomogeneous normal affine $SL_2$-variety. Sbornik. Mathematics, Tome 73 (1992) no. 2, pp. 569-578. http://geodesic.mathdoc.fr/item/SM_1992_73_2_a16/