Line bundles on Bott-Samelson varieties
By N. Lauritzen, J. F. Thomsen. 2004
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Let G be a connected semisimple, simply connected linear algebraic group over an algebraically closed field k and B be a Borel subgroup in G. If w = (P1, . . . , Pn) is a sequence of minimal parabolic subgroups containing B, we may form the quotient Zw = Pw/B , where Pw = P1 × · · · × Pn and B n acts on Pw from the right via (p1, . . . , pn)(b1, . . . , bn) = (p1b1, b −1 1 p2b2, b −1 2 p3b3, . . . , b −1 n−1pnbn). The quotient Zw is inductively a sequence of P -bundles with natural sections starting with the P-bundle P1/B (over a point). The product map Pw → G induces a proper morphism φw : Zw → G/B whose image is a Schubert variety in G/B. For “reduced” sequences w the morphism φw is birational and equal to the celebrated Demazure desingularization of the Schubert variety φw(Zw). In general we call Zw the Bott-Samelson variety associated with w .The construction of Zw originates in the papers [1][2][3] of Bott-Samelson, Demazure and Hansen. See also the master’s thesis (speciale) [4] by Hansen. We characterize the globally generated, ample and very ample line bundles on Zw. The generators of the ample cone are naturally defined O(1)-bundles for successive P-bundles. They form a basis of Pic (Zw). Proving that they account for all ample line bundles originally lead us to some quite involved computer calculations in the Chow ring of Zw. It later turned out that the key point is Lemma 2.1. Using Frobenius splitting [7] and our description of globally generated line bundles we prove the vanishing theorem
Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/d8c53d5cf4ffd1424124d70ac9288dabfc6c0495
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Citations in the corpus
- A survey on the Campana-Peternell Conjecture (2014, methodology): “Following [37], when dealing with finite sequences of indices inD = {1, 2, .”
- Small resolutions of minuscule Schubert varieties (2006, methodology): “We can now describe the ample cone (see also [ LT ]) and the cone of effective curves.”