There is a general problem to describe singularities which can be met on algebraic hypersurfaces, in particular on plane curves, of fixed degree (see, e.g., ). Here we shall consider Ak–singularities which can be met on a plane curve of degree d. Let k(d) be the maximal possible integer k such that there exists a plane curve of degree d with an Ak–singularity. Statement 1 gives an upper bound for k(d). According to it limd→∞k(d)/d 2 ≤ 3/4. We construct a plane curve of degree 28s + 9 (s ∈ Z≥0) which has an Ak– singularity with k = 420s+269s+42. Therefore one has limd→∞k(d)/d 2 ≥ 15/28 (pay attention that 15/28 > 1/2). The example is constructed basically in the same way as a curve of degree 22 with an A257 singularity in  (the aim of that example was somewhat different).