On Milnor classes via invariants of singular subschemes

Theoretical Computer Science

2017

Let V be a possibly singular scheme-theoretic complete intersection subscheme of P n over an algebraically closed field of characteristic zero. Using a recent result of Fullwood ("On Milnor classes via invariants of singular subschemes", Journal of Singularities) we develop an algorithm to compute the Chern-Schwartz-MacPherson class and Euler characteristic of V . This algorithm complements existing algorithms by providing performance improvements in the computation of the Chern-Schwartz-MacPherson class and Euler characteristic for certain types of complete intersection subschemes of P n .

Section: Introductionmentioning

confidence: 99%

“…Note that other sign conventions may be used in definition of the Milnor class, we use the sign convention used by [12], see Fullwood [12] or Aluffi [3] for more details.…”

Section: Introductionmentioning

confidence: 99%

A direct algorithm to compute the topological Euler characteristic and Chern–Schwartz–MacPherson class of projective complete intersection varieties

Theoretical Computer Science

2017

“…where c i is the dimension i component of given by employing Theorem 1.1 of Fullwood [15]. In our case the result of [15,Theorem 1.1] gives the following,…”

Section: The C Sm Class Of Complete Intersectionsmentioning

confidence: 62%

“…. , f r ; this gives (15). Note that while Y ι will depend on the choice of the linear subspace P r−ι (which corresponds to a choice of λ (i) j ∈ k) the class [Y ι ] will not as long as the choice is sufficiently general, i.e.…”

Section: The Segre Class Of Subschemesmentioning

confidence: 99%

Computing characteristic classes of subschemes of smooth toric varieties

2017

Journal of Algebra

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.

“…Specifically, combining the relation (1) with Theorem 1.1 of Fullwood [5] and the expression for the cF J class of a locally complete intersection we obtain the following result giving an expression for cSM (V ) assuming that V can be written as the intersection of j hypersurfaces such that some intersection of j − 1 of the hypersurfaces is smooth (scheme theoretically).…”

Section: The Algorithmmentioning

confidence: 98%

An algorithm to compute certain euler characteristics and Chern-Schwartz-MacPherson classes

Proceedings of the 2014 Symposium on Symbolic-Numeric Computation

2014

Let V be a possibly singular scheme-theoretic global complete intersection subscheme of P n and assume that V can be written as the intersection of j hypersurfaces such that the intersection of j − 1 of the hypersurfaces is smooth (scheme theoretically). Using a result of Fullwood [5] we develop a probabilistic algorithm to compute the Chern-SchwartzMacPherson class (cSM ) and Euler characteristic of V . This algorithm complements existing algorithms by providing performance improvements in the computation of the cSM class and Euler characteristic for schemes having the special structure described above.