Geodesic
On the Finiteness of the Number of Orbits on Quasihomogeneous $(\mathbb C^*)^k\times SL_2(\mathbb C)$-manifolds
Matematičeskie zametki, Tome 81 (2007) no. 5, pp. 766-775

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We obtain an effective criterion for the finiteness of the number of orbits contained in the closure of a given $G$-orbit for the case of a rational linear action of the group $G:=(\mathbb C^*)^k\times SL_2(\mathbb C)$ on a finite-dimensional linear space as well as on the projectivization of such a space.
Keywords: the group $SL_2(\mathbb C)$, rational linear action, character lattice, Borel subgroup, analytic curve, irreducible algebraic variety.
Mots-clés : orbit 
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     author = {E. V. Sharoiko},
     title = {On the {Finiteness} of the {Number} of {Orbits} on {Quasihomogeneous} $(\mathbb C^*)^k\times SL_2(\mathbb C)$-manifolds},
     journal = {Matemati\v{c}eskie zametki},
     pages = {766--775},
     publisher = {mathdoc},
     volume = {81},
     number = {5},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2007_81_5_a14/}
}
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E. V. Sharoiko. On the Finiteness of the Number of Orbits on Quasihomogeneous $(\mathbb C^*)^k\times SL_2(\mathbb C)$-manifolds. Matematičeskie zametki, Tome 81 (2007) no. 5, pp. 766-775. http://geodesic.mathdoc.fr/item/MZM_2007_81_5_a14/