On quotients of real algebraic surfaces in \(\mathbb{C} \mathbb{P}^ 3\)
A real surface in \(\mathbb{C} \mathbb{P}^3\) is defined as the zero set of a homogeneous polynomial \(F\) with real coefficients. The space of such polynomials of degree \(d\) is a real projective space \(\mathbb{R} \mathbb{P}^N\) where \(N = N(d)\). The subfamily of polynomials with singular zero locus form the so called discriminant locus. This is a hypersurface \({\mathcal Z}_d \subset \mathbb{R} \mathbb{P}^N\). The nonsingular polynomials form a disconnected set \({\mathcal N}_d = \mathbb{R} \mathbb{N}^N \setminus {\mathcal Z}_d\). For each \(F \in {\mathcal N}_d\) the surface \(F^{-1}(0)\) is acted upon by the natural antiholomorphic involution given by conjugation. Denote the corresponding quotient by \(X_F\). In this paper the author considers polynomials of degree \(d \leq 4\). Given two such polynomials \(F\), \(G\) lying in different components of \({\mathcal N}_d\), the author explicitly describes the surgeries needed to go from \(X_F\) to \(X_G\). The proof relies on an equivariant Morse theoretic technique and on results from the topology of real algebraic surfaces.
scientific article; zbMATH DE number 824638
scientific article