Real Algebraic Surfaces with Many Handles in $(\mathbb{CP}^1)^3$

Research paper by Renaudineau A.

Indexed on: 03 Mar '18Published on: 17 Dec '16Published in: International Mathematics Research Notices


In this text, we study Viro’s conjecture and related problems for real algebraic surfaces in $(\mathbb{CP}^1)^3$. We construct a counterexample to Viro’s conjecture in tridegree $(4,4,2)$ and a family of real algebraic surfaces of tridegree $(d_1,d_2,2)$ in $(\mathbb{CP}^1)^3$ with asymptotically maximal first Betti number of the real part. To perform such constructions, we consider double covers of blow-ups of $(\mathbb{CP}^1)^2$ and we glue singular curves with special position of the singularities adapting the proof of Shustin’s theorem for gluing singular hypersurfaces.