UBC Theses and Dissertations
Enumerative problems in algebraic geometry motivated from physics Leigh, Oliver
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.
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