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Intersection Theory on Tropical Toric Varieties and Compactifications of Tropical Parameter Spaces

  • We study toric varieties over the tropical semifield. We define tropical cycles inside these toric varieties and extend the stable intersection of tropical cycles in R^n to these toric varieties. In particular, we show that every tropical cycle can be degenerated into a sum of torus-invariant cycles. This allows us to tropicalize algebraic cycles of toric varieties over an algebraically closed field with non-Archimedean valuation. We see that the tropicalization map is a homomorphism on cycles and an isomorphism on cycle classes. Furthermore, we can use projective toric varieties to compactify known tropical varieties and study their combinatorics. We do this for the tropical Grassmannian in the Plücker embedding and compactify the tropical parameter space of rational degree d curves in tropical projective space using Chow quotients of the tropical Grassmannian.
  • Schnitt-Theorie auf tropischen torischen Varietäten und Kompaktifizierungen tropischer Parameterräume

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Author:Henning Meyer
URN (permanent link):urn:nbn:de:hbz:386-kluedo-26372
Advisor:Andreas Gathmann
Document Type:Doctoral Thesis
Language of publication:English
Year of Completion:2011
Year of Publication:2011
Publishing Institute:Technische Universität Kaiserslautern
Granting Institute:Technische Universität Kaiserslautern
Acceptance Date of the Thesis:2011/05/13
Date of the Publication (Server):2011/05/18
Tag:Chow Quotient; Tropical Grassmannian; Tropical Intersection Theory
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
MSC-Classification (mathematics):14-XX ALGEBRAIC GEOMETRY / 14Txx Tropical geometry [See also 12K10, 14M25, 14N10, 52B20] / 14T99 None of the above, but in this section
Licence (German):Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011