Korchagin's third conjecture

TL;DR

This paper investigates the properties of degree nine M-curves with three nests in real projective plane, proving Korchagin's conjecture for curves without jumps and exploring restrictions on complex orientations and isotopy types.

Contribution

It proves Korchagin's third conjecture for curves without jumps and analyzes orientation and isotopy restrictions for specific curve configurations.

Findings

Korchagin's conjecture holds for curves without jumps.

Restrictions on complex orientations for even, even, odd curves with jumps.

Identification of admissible rigid isotopy types.

Abstract

We consider the $M$-curves of degree nine with three nests $1 \langle \alpha_i \rangle, i = 1, 2, 3$ in $\mathbb{R}P^2$. After systematic constructions, Korchagin conjectured that at least two of the $\alpha_i$ must be odd. It was later proved that there is always one odd $\alpha_i$. We say that the curve has a jump in a non-empty oval $O$ if there exist four ovals $A, B, C, D$, with $A$ interior to some other non-empty oval $O'$, $D$ exterior, $B, C$ interior to $O$, such that $B$ and $C$ are separated inside of $O$ by any line passing through $A$ and $D$. In this paper, we prove the conjecture for the curves without jump, and we find restrictions on the complex orientations and rigid isotopy types admissible for the curves even, even, odd with jump.