López Martín, Alberto. Topology and geometry of stable map spaces. 2010, University of Zurich, Faculty of Science.
Abstract
Algebraic stacks were introduced as a solution to the construction of moduli spaces in situations where the theories of varieties and schemes were not good enough. This thesis tackles three different projects concerning the topology and geometry of moduli stacks. In the first chapter of this thesis, we study the enumerative significance of genus-zero Gromov-Witten invariants with a Grassmannian target in terms of rational curves in the Grassmannian. The second of the projects in this thesis is the computation of the Betti numbers of the spaces M 0,0 (G(k, n), d) for d = 2, 3, with G(k, n) the Grassmannian parametrizing k-dimensional vector spaces in V = Cn. The last project in this thesis compares the virtual fundamental classes (introduced by Behrend and Fantechi) of the stack of (g, β, µ)-stable ramified maps log Ug,µ (X, β) and of the stack of (g, β, µ)-log stable ramified maps Ug,µ (X, β), constructed using the machinery described by Kim.
Algebraische Stacks sind als Lösung für die Konstruktion von Modulräumen eingeführt worden, wo Varietäten und Schemata nicht ausreichten. Diese Arbeit besteht aus drei verschiedenen Projekten zur Topologie und Geometrie von algebraischen Stacks. Im ersten Kapitel der Arbeit geht es um Gromov-Witten-Invarianten von Kurven des Geschlechts Null in der Grassmannschen. Wir analisieren ihre enumer- ative Bedeutung bezüglich rationaler Kurven in der Grassmannschen. Der zweite Teil handelt von der Berechnung der Betti-Zahlen der Räume M 0,0 (G(k, n), d) für d = 2, 3, wobei G(k, n) die Grassmannsche ist, welche die ∼ k-dimensionale Untervektorräume in V = Cn parametrisiert. Schliesslich vergleichen wir die virtuellen Fundamentalklassen (eingeführt von Behrend und Fantechi) des Stacks von (g, β, µ)-stabilen verzweigten Abbil- dungen Ug,µ (X, β) und des Stacks von (g, β, µ)-log-stabilen verzweigten Abbillog dungen Ug,µ (X, β), die wie in [Kim] beschrieben konstruiert worden sind.
Abstract
Algebraic stacks were introduced as a solution to the construction of moduli spaces in situations where the theories of varieties and schemes were not good enough. This thesis tackles three different projects concerning the topology and geometry of moduli stacks. In the first chapter of this thesis, we study the enumerative significance of genus-zero Gromov-Witten invariants with a Grassmannian target in terms of rational curves in the Grassmannian. The second of the projects in this thesis is the computation of the Betti numbers of the spaces M 0,0 (G(k, n), d) for d = 2, 3, with G(k, n) the Grassmannian parametrizing k-dimensional vector spaces in V = Cn. The last project in this thesis compares the virtual fundamental classes (introduced by Behrend and Fantechi) of the stack of (g, β, µ)-stable ramified maps log Ug,µ (X, β) and of the stack of (g, β, µ)-log stable ramified maps Ug,µ (X, β), constructed using the machinery described by Kim.
Algebraische Stacks sind als Lösung für die Konstruktion von Modulräumen eingeführt worden, wo Varietäten und Schemata nicht ausreichten. Diese Arbeit besteht aus drei verschiedenen Projekten zur Topologie und Geometrie von algebraischen Stacks. Im ersten Kapitel der Arbeit geht es um Gromov-Witten-Invarianten von Kurven des Geschlechts Null in der Grassmannschen. Wir analisieren ihre enumer- ative Bedeutung bezüglich rationaler Kurven in der Grassmannschen. Der zweite Teil handelt von der Berechnung der Betti-Zahlen der Räume M 0,0 (G(k, n), d) für d = 2, 3, wobei G(k, n) die Grassmannsche ist, welche die ∼ k-dimensionale Untervektorräume in V = Cn parametrisiert. Schliesslich vergleichen wir die virtuellen Fundamentalklassen (eingeführt von Behrend und Fantechi) des Stacks von (g, β, µ)-stabilen verzweigten Abbil- dungen Ug,µ (X, β) und des Stacks von (g, β, µ)-log-stabilen verzweigten Abbillog dungen Ug,µ (X, β), die wie in [Kim] beschrieben konstruiert worden sind.
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