ISBN 9783843913829

978-3-8439-1382-9, Reihe Mathematik

Ngoc Anh Pham
Implementing Intersection Theory

121 Seiten, Dissertation Technische Universität Kaiserslautern (2013), Softcover, A5

Zusammenfassung / Abstract

This thesis deals with implementing some aspects of intersection theory in Sage. More precisely, we proceed along the following lines. Chapter 1 is a review of the relevant material from the literature. In Chapter 2 we introduce the Sage package Schubert3 for computations in intersection theory of smooth varieties, written together with Dang Tuan Hiep. In Schubert3, we deal with abstract varieties, abstract sheaves on abstract varieties, and abstract morphisms between abstract varieties. An abstract variety is a pair consisting of a graded algebra over Q and a certain integral map on it. An abstract morphism between abstract varieties is just a graded algebra homomorphism between graded algebras defining these abstract varieties. Based on this, we implement a number of classical techniques and results in intersection theory, and apply them, for instance, to compute the double point formula, the excess intersection formula, and the Porteous formula. This work is greatly inspired by the Maple package Schubert ([ASK]), written by Sheldon Katz and Stein Arild Strømme, and its Macaulay2 descendent Schubert2 ([GSS+]), written by Daniel R. Grayson, Michael E. Stillman, Stein Arild Strømme, David Eisenbud, and Charley Crissman.