Topological types of p-hyperelliptic real algebraic curves
Given a natural number p, a projective irreducible smooth algebraic curve V defined over \({\mathbb{R}}\) is called p-hyperelliptic if there exists a birational isomorphism \(\phi\) of V, of order 2, such that V/\(\phi\) has genus p. This work is concerned with the existence of such curves according to their genus g and the number k of connected components of V(\({\mathbb{R}})\). We prove that Harnack's condition \(1\leq k\leq g\) is sufficient if \(V\setminus V({\mathbb{R}})\) is connected. In case \(V\setminus V({\mathbb{R}})\) non-connected, the following conditions \(1\leq k\leq g+1\) \((g+k\equiv 1 (2)),\) and either \(k=g+1-2q\) for some q, \(0\leq q\leq p\), or \(\epsilon \leq k\leq 2p+2\) with \(\epsilon =1\) for even p, \(\epsilon =2\) for odd p, are necessary and sufficient for the existence of the curve.
topological types of p-hyperelliptic real algebraic curves
existence of p-hyperelliptic real algebraic curves
scientific article; zbMATH DE number 3950716