Geodesic
Algebraic group actions on normal varieties
Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 78 (2017) no. 1, pp. 101-128

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Let $G$ be a connected algebraic $k$-group acting on a normal $k$-variety, where $k$ is a field. We show that $X$ is covered by open $G$-stable quasi-projective subvarieties; moreover, any such subvariety admits an equivariant embedding into the projectivization of a $G$–linearized vector bundle on an abelian variety, quotient of $G$. This generalizes a classical result of Sumihiro for actions of smooth connected affine algebraic groups. 
@article{MMO_2017_78_1_a4,
     author = {M. Brion},
     title = {Algebraic group actions on normal varieties},
     journal = {Trudy Moskovskogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {101--128},
     publisher = {mathdoc},
     volume = {78},
     number = {1},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/MMO_2017_78_1_a4/}
}
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M. Brion. Algebraic group actions on normal varieties. Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 78 (2017) no. 1, pp. 101-128. http://geodesic.mathdoc.fr/item/MMO_2017_78_1_a4/