Seminar in Real and Complex Geometry

Thursday, January 13, 2022, 16:00-17:30, online (via zoom)

Viatcheslav Kharlamov
(Université de Strasbourg)

On surgery invariant counts in real algebraic geometry


Original Welschinger invariants as well as their various generalizations are very sensitive to the change of topology of the underlying real structure. However, as was later noticed, some combinations of Welschinger invariants may have a stronger invariance property which I call "surgery invariance": the property to be preserved under "wall-crossing" and as a result to be independent on a chosen real structure in a given complex deformation class of varieties under consideration. The starting example is the signed count of real lines on cubic surfaces in accordance with B. Segre's division of such lines in 2 kinds, hyperbolic and elliptic. This example gave rise to the discovery of similar counts on higher dimensional hypersurfaces and complete intersections, and served as one of the roots for a development of an integer valued real Schubert calculus. In this talk (based on a work in progress, joint with Sergey Finashin) I intend to discuss an extension of the above example with real lines on cubic surfaces in a bit different direction: from lines on a cubic surface to lines, and even arbitrary degree rational curves, on other del Pezzo surfaces. Apart of surgery invariance property, the invariants we built also have other remarkable properties, like a "magic" direct relation to Gromov-Witten invariants and surprisingly elementary closed computational formulae.