Geodesic
Desingularizations of Schubert Varieties in Double Grassmannians
Funkcionalʹnyj analiz i ego priloženiâ, Tome 42 (2008) no. 2, pp. 56-67

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Let $X=\operatorname{Gr}(k,V)\times\operatorname{Gr}(l,V)$ be the direct product of two Grassmann varieties of $k$- and $l$-planes in a finite-dimensional vector space $V$, and let $B\subset\operatorname{GL}(V)$ be the isotropy group of a complete flag in $V$. We consider $B$-orbits in $X$, which are an analog of Schubert cells in Grassmannians. We describe this set of orbits combinatorially and construct desingularizations for the closures of these orbits, similar to the Bott–Samelson desingularizations for Schubert varieties.
Keywords: Grassmannian, spherical variety, desingularization. 
@article{FAA_2008_42_2_a6,
     author = {E. Yu. Smirnov},
     title = {Desingularizations of {Schubert} {Varieties} in {Double} {Grassmannians}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {56--67},
     publisher = {mathdoc},
     volume = {42},
     number = {2},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2008_42_2_a6/}
}
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E. Yu. Smirnov. Desingularizations of Schubert Varieties in Double Grassmannians. Funkcionalʹnyj analiz i ego priloženiâ, Tome 42 (2008) no. 2, pp. 56-67. http://geodesic.mathdoc.fr/item/FAA_2008_42_2_a6/