On accumulation points of volumes of log surfaces

摘要

Let C subset of [0,1] be a set satisfying the descending chain condition. We show that every accumulation point of volumes of log canonical surfaces (X, B) with coefficients in C can be realized as the volume of a log canonical surface with big and nef K-X + B and with coefficients in (C) over bar boolean OR {1} in such a way that at least one coefficient lies in Acc(C) boolean OR {1}. As a corollary, if (C) over bar subset of Q, then all accumulation points of volumes are rational numbers. This proves a conjecture of Blache. For the set of standard coefficients C-2 = {1 - 1/n vertical bar n is an element of N} boolean OR {1} we prove that the minimal accumulation point is between 1/(7(2) . 42(2)) and 1/42(2).