Segre classes on smooth projective toric varieties

Abstract

Produced in cooperation with Akademika publishing. The thesis is produced by Akademika publishing merely in connection with the thesis defence. Kindly direct all inquiries regarding the thesis to the copyright holder or the unit which grants the doctorate. Writing this thesis has been a long lasting and challenging task. At the time of its completion it is time to thank everyone who has contributed. First, I would like to thank Professor Ragni Piene for giving me a solid training in mathematics, for introducing me to the puzzling problem of cuspidal curves, and for providing the ideas, suggestions and advice necessary to complete this thesis. I am very grateful for everything that you have helped me achieve the past ten years and for the support and care you have provided along the way. This thesis could not have been written without the encouragement I have received from my colleagues in the algebra group. In particular, I have to thank Nikolay Qviller for inspirational, instructive and productive cooperation. I am very much indebted to Georg Muntingh for interesting discussions and invaluable contributions, for helping me with images, and for proofreading. Moreover, I would like to thank Heidi Camilla Mork and Professor Kristian Ranestad for motivation and support. I am also very grateful to Inger Christin Borge and Arne Bernhard Sletsjøe for always believing in me and for including me in amazing teaching projects. Additionally, I want to thank Jørgen Vold Rennemo and John Christian Ottem for sharing enthusiasm for mathematics and for impressing me on an everyday basis. I also want to thank Professor Keita Tono for helpful explanations, and Torsten Fenske for sending me a copy of his PhD-thesis. My PhD-position has been funded by the Institute of Mathematics, University of Oslo, and I have been associated to the Centre of Mathematics for Applications (CMA). I thank you for giving me the opportunity to write this thesis. Additionally, I want to thank everyone in the administration and IT-drift. Through the years I have met a vast number of fantastic people in the corridors of the Niels Henrik Abel building. In particular, I would like to thank Agnieszka,

Computing characteristic classes of subschemes of smooth toric varieties

“…The algorithm of [30] is a generalization of the previous algorithm of Eklund, Jost and Peterson [12] and works by using appropriate saturations to compute the ideals of certain residual schemes and then computing their multi-degrees. The key advantage of our algorithm (Algorithm 1) likely comes from the result of Theorem 3.5 (combined with Theorem 3.4) since this theorem reduces the problem of computing the Segre class of a subscheme to that of finding the number of solutions to certain zero dimensional polynomial systems; a problem for which there are many effective algorithms.…”

“…In [30] Moe and Qviller give an algorithm to compute the Segre class of subschemes of smooth projective toric varieties. The algorithm of Moe and Qviller [30] is based on a result which gives an expression for the Segre class of a subscheme of a smooth projective toric variety in terms of the classes in the Chow ring of certain residual sets which are computed via saturation.…”

“…The main computational step of the algorithm of Moe and Qviller [30] (and similarly for the algorithm of Eklund, Jost and Peterson [12] for subschemes of P n ) is the computation of the saturations to find the residual sets. This can in practice be a quite computationally expensive procedure.…”

“…This can in practice be a quite computationally expensive procedure. Moe and Qviller [30] describe their algorithm which uses this result to obtain the Segre classes in Section 5 of [30]. Runtime comparisons between the algorithm constructed here and that of [30] are given in §5.…”

“…The main computational cost of the algorithm of Moe and Qviller [30] is the computation of the saturations to find the residual sets. This can, in practice, be a quite computationally expensive procedure.…”