On Fano and weak Fano Bott–Samelson–Demazure–Hansen varieties

On Mori cone of Bott towers

“…By using the results of this article, in [Cha17b] we give some applications to Bott-Samelson-Demazure-Hansen (BSDH) variety, which can be described also as a iterated projective line bundle, by degeneration of this variety to a Bott tower. Precisely, we study Fano, weak Fano, log Fano properties for BSDH varieties (see also [Cha17a]). We obtain some vanishing theorems for the cohomology of tangent bundle (and line bundles) on BSDH varieties (see also [CKP15], [CKP] and [CK17]).…”

A note on toric degeneration of a Bott–Samelson–Demazure–Hansen variety

“…By Lemma 6.1, the toric limit Yw is not weak Fano. Also we can see Z(w) is weak Fano but not Fano (see [Cha17,Theorem 5.3]).…”

“…When G is a simple algebraic group and the expressionw is reduced, Fanoness and weak Fanoness of the BSDH variety Z(w) are considered in [Cha17]. Here we have the following results in Kac-Moody setting.…”