Computing characteristic classes of subschemes of smooth toric varieties

Abstract

Let X Σ be a smooth complete toric variety defined by a fan Σ and let V = V (I) be a subscheme of X Σ defined by an ideal I homogeneous with respect to the grading on the total coordinate ring of X Σ . We show a new expression for the Segre class s(V, X Σ ) 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 Σ is a smooth projective toric variety and dehomogenizing the total homogeneous coordinate ring of X Σ 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 Σ ), the Chern-Schwartz-MacPherson class c SM (V ) and the topological Euler characteristic χ(V ) of V . These algorithms can, in particular, be used for subschemes of any product of projective spaces P n 1 × · · · × 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.

Scattering equations: from projective spaces to tropical grassmannians

“…where we used χ(CP 2 ) = 3. The remaining contribution χ(V ) can be evaluated using the algebraic geometry system Macaulay2 [48] with the package CharacteristicClasses.m2 [49,50] as follows. Let R be the coordinate ring of (CP 2 ) 2 (say over Z/pZ for p=32749) with coordinates r_i for i = 0, 1, .…”

Segre classes and invariants of singular varieties

“…This approach yields an algorithm for computing Chern-Schwartz-MacPherson classes of subschemes of P n and more general varieties, based on the computation of Segre classes (cf. [6], [58], [52], [48], [53]). The current Macaulay2 distribution includes the package CharacteristicClasses [54], by Helmer and Christine Jost, which implements this observation.…”

Segre class computation and practical applications

“…Previous work on computing Segre classes of the form s(X, P n ) in A * (P n ) (i.e., the special case where Y = T = P n ) began with the paper [Alu03] and alternative methods were developed in [EJP13,Hel16]. These methods were generalized to compute s(X, T ) in A * (T ) in [MQ13,Hel17]. In [Har17] the scope was extended to compute the Segre class s(X, Y ) pushed forward to A * (P n ) for X ⊂ Y ⊂ P n and Y a variety.…”

“…developed in [EJP13,Hel16]. These methods were generalized to compute s(X, T ) in A * (T ) in [MQ13,Hel17]. In [Har17] the scope was extended to compute the Segre class s(X, Y ) pushed forward to A * (P n ) for X ⊂ Y ⊂ P n and Y a variety.…”