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April 27, 2018

About an Invariant Signed Count of Real Lines on Real Projective Hypersurfaces

As it was observed a few years ago, there exists a certain signed count of lines that, contrary to the honest “cardinal” count, is independent of the choice of a hypersurface, and by this reason provides, as a consequence, a strong lower bound on the honest count. In this invariant signed count the input of a line is given by its local contribution to the Euler number of a certain auxiliary universal vector bundle. The aim of the talk is to present other, in a sense more geometric, interpretations of the signs involved in the invariant count. In particular, this provides certain generalizations of Segre indices of real lines on cubic surfaces and Welschinger weights of real lines on quintic threefolds. This is a joint work with S.Finashin.