UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

November 10, 2015

939 Evans Hall

5:00PM: Algorithms to Compute Characteristic Classes of Subschemes of Certain Toric Varieties

Martin Helmer

Let X be a complete smooth toric variety where all Cartier divisors in Pic(X) are nef and let V be a subscheme of X. We give a new expression for the Segre class of the subscheme V, s(V,X), in terms of the projective degrees of a rational map associated to V. We also give a concrete and computable expression for these projective degrees. These results are applied to develop effective algorithms for the computation of the Chern-Schwartz-MacPherson class, Segre class and the Euler characteristic of V. The algorithms will, in particular, be applicable to any subscheme of a product of projective spaces. In the case of smooth subschemes V this will also allow us to compute the total Chern class of V. The algorithms may be implemented symbolically using Groebner basis or numerically using homotopy continuation via a package such as Bertini or PHCPack. The algorithms have been implemented in Macaulay2 and an M2 package is available. The algorithms described perform favourably on a wide selection of examples in comparison to other known algorithms. Theoretical running time bounds for several of the algorithms are also given. In the talk we will focus on describing the algorithm in the special case where X is a projective space of dimension n.

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