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Enumerative problems in algebraic geometry motivated from physics

University of British Columbia

This thesis contains two chapters which reflect the two main viewpoints of modern enumerative geometry. In chapter 1 we develop a theory for stable maps to curves with divisible ramification. For a fixed integer r>0, we show that the condition of every ramification locus being divisible by r is equivalent to the existence of an r-th root of a canonical section. We consider this condition in regards to both absolute and relative stable maps and construct natural moduli spaces in these situations. We construct an analogue of the Fantechi-Pandharipande branch morphism and when the domain curves are genus zero we construct a virtual fundamental class. This theory is anticipated to have applications to r-spin Hurwitz theory. In particular it is expected to provide a proof of the r-spin ELSV formula [SSZ'15, Conj. 1.4] when used with virtual localisation. In chapter 2 we further the study of the Donaldson-Thomas theory of the banana threefolds which were recently discovered and studied in [Bryan'19]. These are smooth proper Calabi-Yau threefolds which are fibred by Abelian surfaces such that the singular locus of a singular fibre is a non-normal toric curve known as a “banana configuration”. In [Bryan'19] the Donaldson-Thomas partition function for the rank 3 sub-lattice generated by the banana configurations is calculated. In this chapter we provide calculations with a view towards the rank 4 sub-lattice generated by a section and the banana configurations. We relate the findings to the KKV formula and present new Gopakumar-Vafa invariants for the banana threefold.

Text

2019-07-04

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/70908

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/