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# 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

#### 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

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