Bigq-ample line bundles

Abstract

AbstractA recent paper of Totaro developed a theory ofq-ample bundles in characteristic 0. Specifically, a line bundleLonXisq-ample if for every coherent sheaf ℱ onX, there exists an integerm0such thatm≥m0impliesHi(X,ℱ⊗𝒪(mL))=0 fori>q. We show that a line bundleLon a complex projective schemeXisq-ample if and only if the restriction ofLto its augmented base locus isq-ample. In particular, whenXis a variety andLis big but fails to beq-ample, then there exists a codimension-one subschemeDofXsuch that the restriction ofLtoDis notq-ample.