PreprintCC BY 4.0arXiv
On arrangements of plane real quartics with respect to three lines
Abstract
We complete the classification of mutual arrangements of a smooth real algebraic or real pseudoholomorphic quartic curve and three lines under condition that each oval of the quartic intersects the union of the lines. This classification was started in a recent preprint by Maletto. There is one arrangement which is realizable pseudoholomorphically but not algebraically. It can be constructed in different ways, in particular, by a combinatorial patchworking on an irregular triangulation. This is the first example of a combinatorial patchworking which produces a PL curve in RP2 whose arrangement relative to the coordinate axes is algebraically unrealizable.
Related
- Third-Order Weierstrass Points on Plane Quartics
- Seshadri constants of singular loci of line arrangements with at most twelve lines
- Terao's freeness conjecture for arrangements of 15 and 16 lines
- Point-balanced arrangements in the real projective plane and the Hirzebruch property
- Non-simple Arrangements of Lines and Pseudolines with Many Triangles
BibTeX
@misc{orevkov2026arrangements,
title = {{On arrangements of plane real quartics with respect to three lines}},
author = {S. Yu. Orevkov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.19457},
eprint = {2607.19457},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.19457},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On arrangements of plane real quartics with respect to three lines
AU - S. Yu. Orevkov
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2607.19457
UR - https://arxiv.org/abs/2607.19457
AB - We complete the classification of mutual arrangements of a smooth real algebraic or real pseudoholomorphic quartic curve and three lines under condition that each oval of the quartic intersects the union of the lines. This classification was started in a recent preprint by Maletto. There is one arrangement which is realizable pseudoholomorphically but not algebraically. It can be constructed in different ways, in particular, by a combinatorial patchworking on an irregular triangulation. This is the first example of a combinatorial patchworking which produces a PL curve in RP^2 whose arrangement relative to the coordinate axes is algebraically unrealizable.
ER -