Frobenius Splittings and Blow-ups

Maximal compatible splitting and diagonals of Kempf varieties

“…In the last part ( §6) of this paper, we enhance the geometric arguments in [11] and show that the Gaussian map (1.1) is surjective, provided that L 1 = L⊗M 1 and L 2 = L⊗M 2 , where L is ample and M 1 , M 2 globally generated line bundles on X (a projective smooth variety) and the diagonal ∆ ⊂ X × X is maximally compatibly split. Here we do not need the underlying field to have odd characteristic (as in [11]).…”

“…Here we do not need the underlying field to have odd characteristic (as in [11]). This enables us to prove Wahl's conjecture also for Kempf varieties, since they posses unique minimal ample line bundles as Schubert varieties in G/B.…”

“…In [11] Let X = G/P , where G is a semisimple linear algebraic group and P ⊂ G a parabolic subgroup. In a beautiful geometric argument Lakshmibai, Mehta and Parameswaran proved that a Frobenius splitting of the blow-up Bl ∆ (X × X) compatibly splitting the exceptional divisor implies Wahl's conjecture in positive characteristic.…”

“…They conjectured the existence of a Frobenius splitting of X × X vanishing with maximal multiplicity on the diagonal ∆ (we refer to this as the LMP conjecture, cf. §2.4 in [11] and §2.C in [2]). …”

Spherical varieties and Wahl’s conjecture

“…This conjecture was proved by Kumar in characteristic 0 in [10]. Lakshmibai, Mehta and Parameswaran [11] considered the situation in positive characteristic and proved that the following conjecture (now called LMPconjecture) implies Wahl's conjecture in positive characteristic. From now on in the introduction, the base field k is algebraically closed of positive characteristic p.…”