In this note, we construct a sequence of l.t. surfaces (Xn)n ∈ ℕ such that KXn is ample for all n and such that (K2Xn)n ∈ ℕ is a strictly increasing series with limit equal to 1. This answers (in the affirmative) a question by Alexeev, cf. [Al], 11·1. Here, an l.t. surface is a normal complex projective surface with at most quotient singularities (which is the same as ‘at most log terminal singularities’). A main result of [Al] implies that it is impossible to find a sequence (Xn)n ∈ ℕ of l.t. surfaces with KXn ample for all n such that K2Xn is strictly decreasing. Although our construction is not too difficult, the example is new and has several interesting implications, see Section 4.

Without further explanation, we use some fundamental tools concerning l.t. surfaces like Mumford's intersection theory or the notion of minimality; the reader should consult [Blb] and the references quoted there.