Segre Indices and Welschinger Weights as Options for Invariant Count of Real Lines

Abstract

Abstract In our previous paper [5] we have elaborated a certain signed count of real lines on real hypersurfaces of degree $2n-1$ in $P^{n+1}$. Contrary to the honest “cardinal” count, it is independent of the choice of a hypersurface and, by this reason, provides a strong lower bound on the honest count. In this count the contribution of a line is its local input to the Euler number of a certain auxiliary vector bundle. The aim of this paper is to present other, in a sense more geometric, interpretations of this local input. One of them results from a generalization of Segre species of real lines on cubic surfaces and another from a generalization of Welschinger weights of real lines on quintic three-folds.