Moscow Mathematical Journal

Volume 22, Issue 3, July–September 2022  pp. 427–450.

On the Bounded Negativity Conjecture and Singular Plane Curves

Authors:  Alexandru Dimca (1), Brian Harbourne (2), and Gabriel Sticlaru (3)
Author institution:(1) Université Côte d'Azur, CNRS, LJAD, France and Simion Stoilow Institute of Mathematics, P.O. Box 1-764, RO-014700 Bucharest, Romania
(2) Math Department, University of Nebraska–Lincoln, Lincoln, NE 68588 USA
(3) Faculty of Mathematics and Informatics, Ovidius University, Bd. Mamaia 124, 900527 Constanta, Romania

Summary:

There are no known failures of Bounded Negativity in characteristic 0. In the light of recent work showing the Bounded Negativity Conjecture fails in positive characteristics for rational surfaces, we propose new characteristic free conjectures as a replacement. We also develop bounds on numerical characteristics of curves constraining their negativity. For example, we show that the $H$-constant of a rational curve $C$ with at most 9 singular points satisfies $H(C)>-2$ regardless of the characteristic.

2020 Math. Subj. Class. Primary: 14H50; Secondary: 14B05, 32S05.

Keywords: Bounded negativity conjecture, plane curves, singularities, rational curves, ordinary singularities.