Let X be a smooth complete Fano variety over C. We show that the Cox ring L∈Pic(X) H 0 (X, OX (L)) is Gorenstein with canonical singularities.

We prove a conjecture of Shokurov which characterises toric varieties using log pairs.

Let Γ be a finite graph and let Γ n be the "nth cone over Γ" (i.e., the join of Γ and the complete graph K n ). We study the asymptotic structure of the chip-firing group Pic 0 (Γ n ).

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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.