An Algorithm to Compute the Topological Euler Characteristic, the Chern-Schwartz-MacPherson Class and the Segre class of Subschemes of Some Smooth Complete Toric Varieties

Let $X_{\Sigma}$ be a complete smooth toric variety of dimension $n$ defined by a fan $\Sigma$ where all Cartier divisors in $\mathrm{Pic}(X_{\Sigma})$ are nef and let $V$ be a subscheme of $X_{\Sigma}$. We show a new expression for the Segre class $s(V,X_{\Sigma})$ 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 $\Sigma$ is equal to the number of generating rays in $\Sigma(1)$ minus the dimension of $X_{\Sigma}$ 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 $s(V,X_{\Sigma})$, the Chern-Schwartz-MacPherson class $c_{SM}(V)$ and the Euler characteristic $\chi(V)$ 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 $\mathbb{P}^{n_1} \times \cdots \times \mathbb{P}^{n_j}$ (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 examples. In all cases the algorithm to compute the Segre class is found to offer significantly increased performance over other known algorithms. At present we know of no other algorithms which compute Chern-Schwartz-MacPherson classes and Euler characteristics in this setting.