On arrangements of plane real quartics with respect to three lines

2026/07/21 by S. Yu. Orevkov
#math.AG

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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.

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