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Intersection theory with applications to the computation of Gromov-Witten invariants

  • This thesis is devoted to the computational aspects of intersection theory and enumerative geometry. The first results are a Sage package Schubert3 and a Singular library schubert.lib which both provide the key functionality necessary for computations in intersection theory and enumerative geometry. In particular, we describe an alternative method for computations in Schubert calculus via equivariant intersection theory. More concretely, we propose an explicit formula for computing the degree of Fano schemes of linear subspaces on hypersurfaces. As a special case, we also obtain an explicit formula for computing the number of linear subspaces on a general hypersurface when this number is finite. This leads to a much better performance than classical Schubert calculus. Another result of this thesis is related to the computation of Gromov-Witten invariants. The most powerful method for computing Gromov-Witten invariants is the localization of moduli spaces of stable maps. This method was introduced by Kontsevich in 1995. It allows us to compute Gromov-Witten invariants via Bott's formula. As an insightful application, we computed the numbers of rational curves on general complete intersection Calabi-Yau threefolds in projective spaces up to degree six. The results are all in agreement with predictions made from mirror symmetry.

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Author:Hiep Dang
URN (permanent link):urn:nbn:de:hbz:386-kluedo-37506
Advisor:Wolfram Decker
Document Type:Doctoral Thesis
Language of publication:English
Publication Date:2014/02/03
Year of Publication:2014
Publishing Institute:Technische Universität Kaiserslautern
Granting Institute:Technische Universität Kaiserslautern
Acceptance Date of the Thesis:2014/01/24
Date of the Publication (Server):2014/03/14
Number of page:VII, 125
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
MSC-Classification (mathematics):14-XX ALGEBRAIC GEOMETRY / 14Cxx Cycles and subschemes / 14C17 Intersection theory, characteristic classes, intersection multiplicities [See also 13H15]
14-XX ALGEBRAIC GEOMETRY / 14Nxx Projective and enumerative geometry [See also 51-XX] / 14N15 Classical problems, Schubert calculus
14-XX ALGEBRAIC GEOMETRY / 14Nxx Projective and enumerative geometry [See also 51-XX] / 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
14-XX ALGEBRAIC GEOMETRY / 14Qxx Computational aspects in algebraic geometry [See also 12Y05, 13Pxx, 68W30] / 14Q15 Higher-dimensional varieties
Licence (German):Standard gemäß KLUEDO-Leitlinien vom 10.09.2012