Real algebraic curves of bidegree $(5,5)$ on the quadric ellipsoid

In this paper, the topological classification of nonseparating (respectively, separating) real algebraic nonsingular $(M-i)$-curves of bidegree $ (5, 5)$ on the quadric ellipsoid is completed. In particular, it is shown that previously known restrictions form a complete system for this bidegree. Therefore, the main part of the paper concerns the construction of real algebraic curves. The strategy is to reduce the problem of construction of curves on the quadric ellipsoid to construction of curves on the second Hirzebruch surface by degenerating the quadric ellipsoid to the quadratic cone. Next, various classical construction methods on toric surfaces are combined, such as dessins d'enfants and Viro's patchworking method.