10 research outputs found
The Euclidean distance degree of smooth complex projective varieties
We obtain several formulas for the Euclidean distance degree (ED degree) of
an arbitrary nonsingular variety in projective space: in terms of Chern and
Segre classes, Milnor classes, Chern-Schwartz-MacPherson classes, and an
extremely simple formula equating the Euclidean distance degree of X with the
Euler characteristic of an open subset of X
Segre Class Computation and Practical Applications
Let be closed (possibly singular) subschemes of a smooth
projective toric variety . We show how to compute the Segre class
as a class in the Chow group of . Building on this, we give effective
methods to compute intersection products in projective varieties, to determine
algebraic multiplicity without working in local rings, and to test pairwise
containment of subvarieties of . Our methods may be implemented without
using Groebner bases; in particular any algorithm to compute the number of
solutions of a zero-dimensional polynomial system may be used
Toric polar maps and characteristic classes
Given a hypersurface in the complex projective space, we prove that the
degree of its toric polar map is given by the signed topological Euler
characteristic of a distinguished open set, namely the complement of the union
of the hypersurface and the coordinate hyperplanes. In addition, we prove that
if the hypersurface is in general position or is nondegenerate with respect to
its Newton polytope, then the coefficients of the Chern-Schwartz-MacPherson
class of the distinguished open set agree, up to sign, with the multidegrees of
the toric polar map. In the latter case, we also recover the multidegrees from
mixed volumes.
For plane curves, a precise formula for the degree of the toric polar map is
obtained in terms of local invariants. Finally, we construct families, in
arbitrary dimension, of irreducible hypersurfaces whose toric polar map is
birational.Comment: v2: Fixed typos, minor improvements. 28 pages. Comments welcome