Doctor of Philosophy
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.
Helmer, Martin, "Algorithms to Compute Characteristic Classes" (2015). Electronic Thesis and Dissertation Repository. 2923.