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.
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