TY - JOUR

T1 - Computing characteristic classes of subschemes of smooth toric varieties

AU - Helmer, Martin

PY - 2017/4/15

Y1 - 2017/4/15

N2 - Let XΣ be a smooth complete toric variety defined by a fan Σ and let V=V(I) be a subscheme of XΣ defined by an ideal I homogeneous with respect to the grading on the total coordinate ring of XΣ. We show a new expression for the Segre class s(V,XΣ) 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 XΣ is a smooth projective toric variety and dehomogenizing the total homogeneous coordinate ring of XΣ 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,XΣ), the Chern–Schwartz–MacPherson class cSM(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 Pn1 ×⋯×Pnj 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.

AB - Let XΣ be a smooth complete toric variety defined by a fan Σ and let V=V(I) be a subscheme of XΣ defined by an ideal I homogeneous with respect to the grading on the total coordinate ring of XΣ. We show a new expression for the Segre class s(V,XΣ) 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 XΣ is a smooth projective toric variety and dehomogenizing the total homogeneous coordinate ring of XΣ 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,XΣ), the Chern–Schwartz–MacPherson class cSM(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 Pn1 ×⋯×Pnj 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.

KW - Chern class

KW - Chern–Schwartz–MacPherson class

KW - Computational intersection theory

KW - Computer algebra

KW - Euler characteristic

KW - Segre class

KW - Toric varieties

UR - http://www.scopus.com/inward/record.url?scp=85010961150&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2016.12.024

DO - 10.1016/j.jalgebra.2016.12.024

M3 - Journal article

AN - SCOPUS:85010961150

VL - 476

SP - 548

EP - 582

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -