Geodesic
Multiple Flag Varieties
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Seminar on algebra and geometry of the Samara University, Tome 147 (2018), pp. 84-119

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This paper is a review of results on multiple flag varieties, i.e., varieties of the form $G/P_1\times\dots\times G/P_r$. We provide a classification of multiple flag varieties of complexity $0$ and $1$ and results on the combinatorics and geometry of $B$-orbits and their closures in double cominuscule flag varieties. We also discuss questions of finiteness for the number of $G$-orbits and existence of an open $G$-orbits on a multiple flag variety.
Keywords: flag varieties, spherical varieties. 
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     author = {E. Yu. Smirnov},
     title = {Multiple {Flag} {Varieties}},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {84--119},
     publisher = {mathdoc},
     volume = {147},
     year = {2018},
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E. Yu. Smirnov. Multiple Flag Varieties. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Seminar on algebra and geometry of the Samara University, Tome 147 (2018), pp. 84-119. http://geodesic.mathdoc.fr/item/INTO_2018_147_a2/