Some Examples of Real Algebraic and Real Pseudoholomorphic Curves

In this paper we construct several examples (series of examples) of real algebraic and real pseudoholomorphic curves in RP2 in which we tried to maximize different characteristics among curves of a given degree. In Sect. 2, this is the number of nonempty ovals; in Sect. 4, the number of ovals of the maximal depth; in Sect. 5, the number n such that the curve has an An singularity. In the pseudoholomorphic case, the questions of Sects. 4 and 5 are equivalent to the same problem about braids, which is studied in Sect. 3. In Sect. 6.1, we construct a real algebraic M-curve of degree 4d+1 with four nests of depth d (which shows that the congruence mod 8 proven in a joint paper with Viro is “nonempty”). In Sect. 6.2, we generalize this construction. In Sect. 7, we construct real algebraic M-curves of degree 9 with a single exterior oval, and we classify such curves up to isotopy.