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Real algebraic curves on real minimal del Pezzo surfaces

Banff International Research Station for Mathematical Innovation and Discovery

The study of the topology of real algebraic varieties dates back to the work of Harnack, Klein and Hilbert in the 19th century; in particular, the isotopy type classification of real algebraic curves with a fixed degree in the real projective plane is a classical subject that has undergone considerable evolution. On the other hand, apart from studies concerning Hirzebruch surfaces and at most degree 3 surfaces in the real projective space, not much is known for more general ambient surfaces. In particular, this is because varieties constructed using the patchworking method are hypersurfaces of toric varieties. However, there are many other real algebraic surfaces. Among these are the real rational surfaces, and more particularly the real minimal rational surfaces. In this talk, we present some results about the classification of topological types realized by real algebraic curves of "small class" in real minimal del Pezzo surfaces which are real non-toric surfaces with non-connected real parts. We will explain how combine variations of classical methods with degeneration methods, that have found recent applications in real enumerative geometry, and the exploitation of Welschinger invariants to get through such classifications.

Mathematics; Algebraic geometry; Combinatorics

video/mp4

Author affiliation: Ecole Polytechnique

2020-03-09

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/73703

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/