  This paper reveals some new analytical and geometrical properties of the
generalized algebraic multiplicity, $\chi$, introduced in [7, 5] and further
developed in [20, 23, 24]. In particular, it establishes a completely new
connection between $\chi$ and the concept of local intersection index of
algebraic varieties, a central device in Algebraic Geometry. This link between
Nonlinear Spectral Theory and Algebraic Geometry provides to $\chi$ with a deep
geometrical meaning. Moreover, $\chi$ is characterized through the new notion
of local determinant of the Schur operator associated to the linear path,
$\mathfrak{L}(\lambda)$.
