The double of the doubles of Klein surfaces

A Klein surface is a surface with a dianalytic structure. A double of a Klein surface $X$ is a Klein surface $Y$ such that there is a degree two morphism (of Klein surfaces) $Y\rightarrow X$. There are many doubles of a given Klein surface and among them the so-called natural doubles which are: the complex double, the Schottky double and the orienting double. We prove that if $X$ is a non-orientable Klein surface with non-empty boundary, the three natural doubles, although distinct Klein surfaces, share a common double: "the double of doubles" denoted by $DX$. We describe how to use the double of doubles in the study of both moduli spaces and automorphisms of Klein surfaces. Furthermore, we show that the morphism from $DX$ to $X$ is not given by the action of an isometry group on classical surfaces.Comment: 14 pages; more details in the proof of theorem

Note on the degree of C°-sufficiency of plane curves

Let f be a germ of plane curve, we define the 8-degree of sufficiency offto be the smallest integer r such that for any germ g such that j(r)f = j(r)g then there is a set of disjoint annuli in S3 whose boundaries consist of a component of the link of f and a component of the link of g. We establish a formula for the 8-degree of sufficiency in terms of link invariants of plane curves singularities and, as a consequence of this formula, we obtain that the 8-degree of sufficiency is equal to the C° -degree of sufficienc

On two recent geometrical characterizations of hyperellipticity

We obtain short and unified new proofs of two recent characterizations of hyperellipticity given in [4] and [6], as well as a way of establishing a relation between them