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SCHUBERT VARIETIES ARE LOG FANO DAVE ANDERSON AND ALAN STAPLEDON
 

Summary: SCHUBERT VARIETIES ARE LOG FANO
DAVE ANDERSON AND ALAN STAPLEDON
Abstract. For a Schubert variety Xw, we exhibit a divisor such that the
pair (Xw, ) is log Fano, independent of the characteristic.
Let Xw = BwB/B be a Schubert variety in a Kac-Moody flag variety G/B over
an algebraically closed field of arbitrary characteristic. Fix a reduced word for the
Weyl group element w, and let : Xw Xw be the corresponding Bott-Samelson
resolution. Let = 1 + + k be the complement of the open B-orbit in Xw,
written as a sum of prime divisors, and define = 1 + + similarly; the divisor
is a simple normal crossings divisor. Here = (w) is equal to the length of w,
which in turn is equal to the dimension of Xw.
Denote by the sum of all fundamental weights of (the simply connected form
of) G, and let L() be the corresponding ample line bundle on Xw. Choosing
B-invariant sections of L() and L(), these line bundles correspond to divisors
a11 + + akk, and b11 + + b,
for some nonnegative integers ai and bi. In fact, they are all positive, by the Lemma
below.
Recall that for a normal irreducible variety Y and an effective Q-divisor D, the
pair (Y, D) is Kawamata log terminal (klt) if KY + D is Q-Cartier, and for
all proper birational maps : Y Y , the pullback (KY + D) = KY + D has

  

Source: Anderson, Dave - Department of Mathematics, University of Washington at Seattle

 

Collections: Mathematics