Computing characteristic classes of subschemes of smooth toric varieties

Let XΣ be a complete smooth toric variety of dimension n defined by a fan Σ where all Cartier divisors in Pic(XΣ) are nef and let V be a subscheme of XΣ. We show a new expression for the Segre class s(V,XΣ) in terms of the projective degrees of a rational map associated to V . In the case where the number of primitive collections of rays in the fan Σ is equal to the number of generating rays in Σ(1) minus the dimension of XΣ we give an explicit expression for the projective degrees which can be easily computed using a computer algebra system. We apply this to give effective algorithms to compute the Segre class s(V,XΣ), the Chern-Schwartz-MacPherson class cSM (V ) and the Euler characteristic χ(V ) of V . These algorithms can, in particular, compute the Segre class, Chern-Schwartz-MacPherson class and Euler characteristic of arbitrary subschemes of any product of projective spaces Pn1 × · · · × Pnj (over an algebraically closed field of characteristic zero). Running time bounds for several of the algorithms are given and the algorithms are tested on a variety of