Computing characteristic classes of subschemes of smooth toric varieties

Abstract

Let $X_{\Sigma}$ be a smooth complete toric variety defined by a fan $\Sigma$ and let $V=V(I)$ be a subscheme of $X_{\Sigma}$ defined by an ideal $I$ homogeneous with respect to the grading on the total coordinate ring 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 specified by the generators of $I$ when each generator corresponds to a numerically effective (nef) divisor. Restricting to the case where $X_{\Sigma}$ is a smooth projective toric variety and dehomogenizing the total homogeneous coordinate ring of $X_{\Sigma}$ via a dehomogenizing ideal we also give an expression for the projective degrees of this rational map in terms of the dimension of an explicit quotient ring. Under an additional technical assumption we construct what we call a general dehomogenizing ideal and apply this construction to give effective algorithms to compute the Segre class $s(V,X_{\Sigma})$, the Chern-Schwartz-MacPherson class $c_{SM}(V)$ and the topological Euler characteristic $\chi(V)$ of $V$. These algorithms can, in particular, be used for subschemes of any product of projective spaces $\mathbb{P}^{n_1} \times \cdots \times \mathbb{P}^{n_j}$ or for subschemes of many other projective toric varieties. Running time bounds for several of the algorithms are given and the algorithms are tested on a variety of examples. In all applicable cases our algorithms to compute these characteristic classes are found to offer significantly increased performance over other known algorithms.