On the Geometry of the Anti-canonical Bundle of the Bott–Samelson–Demazure–Hansen Varieties

Let G be a semi-simple simply connected algebraic group over the field C of complex numbers.Let T be a maximal torus of G,and let W be the Weyl group of G with respect to T.Let Z(w,i)be the Bott–Samelson–Demazure–Hansen variety corresponding to a tuple i associated to a reduced expression of an element w∈W.We prove that for the tuple i associated to any reduced expression of a minuscule Weyl group element w,the anti-canonical line bundle on Z(w,i)is globally generated.As consequence,we prove that Z(w,i)is weak Fano.Assume that G is a simple algebraic group whose type is different from A2.Let S={α1,...,αn}be the set of simple roots.Let w be such that support of w is equal to S.We prove that Z(w,i)is Fano for the tuple i associated to any reduced expression of w if and only if w is a Coxeter element and w^(−1)(Σ_(t=1)^(n)α_(t))∈−S.