Algorithms to Compute Characteristic Classes

In this thesis we develop several new algorithms to compute characteristics classes in a variety of settings. In addition to algorithms for the computation of the Euler characteristic, a classical topological invariant, we also give algorithms to compute the Segre class and Chern-Schwartz-MacPherson (CSM) class. These invariants can in turn be used to compute other common invariants such as the Chern-Fulton class (or the Chern class in smooth cases). We begin with subschemes of a projective space over an algebraically closed field of characteristic zero. In this setting we give effective algorithms to compute the CSM class, Segre class and the Euler characteristic. The algorithms can be implemented using either symbolic or numerical methods. The algorithms are based on a new method for calculating the projective degrees of a rational map defined by a homogeneous ideal. Running time bounds are given for these algorithms and the algorithms are found to perform favourably compared to other applicable algorithms. Relations between our algorithms and other existing algorithms are explored. In the special case of a complete intersection subcheme we develop a second algorithm to compute CSM classes and Euler characteristics in a more direct and efficient manner. Each of these algorithms are generalized to subschemes of a product of projective spaces. Running time bounds for the generalized algorithms to compute the CSM class, Segre class and the Euler characteristic are given. Our Segre class algorithm is tested in comparison to another applicable algorithm and is found to perform favourably. To the best of our knowledge there are no other algorithms in the literature which compute the CSM class and Euler characteristic in the multi-projective setting. For complete simplical toric varieties defined by a fan we give a strictly combinatorial algorithm to compute the CSM class and Euler characteristic and a second combinatorial algorithm with reduced running time to compute only the Euler characteristic. We also prove several Bezout type bounds in multi-projective space. An application of these bounds to obtain a sharper degree bound on a certain system with a natural bi-projective structure is demonstrated

An Algorithm to Compute the Topological Euler Characteristic, Chern-Schwartz-MacPherson Class and Segre Class of Projective Varieties

Let $V$ be a closed subscheme of a projective space $\mathbb{P}^n$. We give an algorithm to compute the Chern-Schwartz-MacPherson class, Euler characteristic and Segre class of $V$. The algorithm can be implemented using either symbolic or numerical methods. The algorithm is based on a new method for calculating the projective degrees of a rational map defined by a homogeneous ideal. Using this result and known formulas for the Chern-Schwartz-MacPherson class of a projective hypersurface and the Segre class of a projective variety in terms of the projective degrees of certain rational maps we give algorithms to compute the Chern-Schwartz-MacPherson class and Segre class of a projective variety. Since the Euler characteristic of $V$ is the degree of the zero dimensional component of the Chern-Schwartz-MacPherson class of $V$ our algorithm also computes the Euler characteristic $\chi(V)$. Relationships between the algorithm developed here and other existing algorithms are discussed. The algorithm is tested on several examples and performs favourably compared to current algorithms for computing Chern-Schwartz-MacPherson classes, Segre classes and Euler characteristics

A Macaulay2 package for characteristic classes and the topological Euler characteristic of complex projective schemes

The Macaulay2 package CharacteristicClasses provides commands for the computation of the topological Euler characteristic, the degrees of the Chern classes and the degrees of the Segre classes of a closed subscheme of complex projective space. The computations can be done both symbolically and numerically, the latter using an interface to Bertini. We provide some background of the implementation and show how to use the package with the help of examples.Comment: 6 page

The Euclidean distance degree of smooth complex projective varieties

We obtain several formulas for the Euclidean distance degree (ED degree) of an arbitrary nonsingular variety in projective space: in terms of Chern and Segre classes, Milnor classes, Chern-Schwartz-MacPherson classes, and an extremely simple formula equating the Euclidean distance degree of X with the Euler characteristic of an open subset of X

A method to compute Segre classes of subschemes of projective space

We present a method to compute the degrees of the Segre classes of a subscheme of complex projective space. The method is based on generic residuation and intersection theory. It has been implemented using the software system Macaulay2.Comment: 13 page