On curves of degree 10 with 12 triple points

Abstract: We construct an irreducible rational curve of degree 10 in $CP^2$ which has 12 triple points. This gives a counter-example to a conjecture by Dimca, Harbourne, and Sticlaru. We also prove that there exists an analytic family $C_u$ of such curves which converges as $u\to 0$ to the union of the dual Hesse arrangement of lines (9 lines with 12 triple points) with an additional line. We hope that our approach to the proof of the latter fact could be of independent interest.