- Published 1999

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., [2]). 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 [3] (the aim of that example was somewhat different).

@inproceedings{Nekhoroshev1999SMGuseinZade,
title={S.M.Gusein–Zade ∗},
author={N . N . Nekhoroshev},
year={1999}
}