Chambers and walls in spaces of real algebraic curves of small degrees

Abstract

This paper reviews known results on the rigid isotopy classification of plane curves of degree $m\leq6$ and curves of small degrees on quadrics. The paper's study completes the rigid isotopy classification of nonsingular real algebraic curves of bidegree (4,3) on a hyperboloid, begun by the author in earlier works. There are given previously missing proofs of the uniqueness of the connected components for 16 classes of real algebraic curves of bidegree (4,3) having a single node or a cusp. The main technical tools are graphs of real trigonal curves on Hirzebruch surfaces. Adjacency graphs of chambers and walls in the spaces of these curves are presented.