Methods and Applications of Analysis

Volume 24 (2017)

Number 2

Special issue dedicated to Henry B. Laufer on the occasion of his 70th birthday: Part 2

Guest Editors: Stephen S.-T. Yau (Tsinghua University, China); Gert-Martin Greuel (University of Kaiserslautern, Germany); Jonathan Wahl (University of North Carolina, USA); Rong Du (East China Normal University, China); Yun Gao (Shanghai Jiao Tong University, China); and Huaiqing Zuo (Tsinghua University, China)

Sextic curves with six double points on a conic

Pages: 295 – 302

DOI: http://dx.doi.org/10.4310/MAA.2017.v24.n2.a6

Authors

Kazuhiro Konno (Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, Japan)

Ezio Stagnaro (Dipartimento di Tecnica e Gestione dei Sistemi Industriali, Università di Padova, Vicenza, Italy)

Abstract

Let $C_6$ be a plane sextic curve with $6$ double points that are not nodes. It is shown that if they are on a conic $C_2$, then the unique possible case is that all of them are ordinary cusps. From this it follows that $C_6$ is irreducible. Moreover, there is a plane cubic curve $C_3$ such that $C_6 = C^3_2 + C^2_3$. Such curves are closely related to both the branch curve of the projection to a plane of the general cubic surface from a point outside it and canonical surfaces in $\mathbb{P}^3$ or $\mathbb{P}^4$ whose desingularizations have birational invariants $q \gt 0, p_g = 4$ or $p_g = 5, P_2 \leq 23$.

Keywords

plane curves, ordinary cusps, tacnodes, surfaces of general type

2010 Mathematics Subject Classification

14H20, 14H45, 14H50, 14J25, 14J29

Full Text (PDF format)

The first named author is partially supported by Grants-in-Aid for Scientific Research (A) (No. 24244002) by Japan Society for the Promotion of Science (JSPS).

Received 31 October 2016

Accepted 29 July 2017

Published 3 January 2018