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Virtual invariants on Quot Schemes over Fano surfaces


Moduli theory, a subfield of algebraic geometry, focuses on computing geometric enumerative invariants. Building on earlier classical methods, algebraic geometers in the 20th century began approaching these questions using intersection theory, allowing many new and interesting examples. The primary focus of this paper is the computation of several (virtual) enumerative invariants on the Quot scheme of rank N-r quotients of On/s with Euler characteristic [chi] and first Chern class [beta] over a Fano surface S, for which we use the shorthand notation Quots. This project is a natural generalization of the work of Aaron Bertram, Alina Marian, Dragos Oprea, and others, who considered the analogous problem on a Quot scheme over a smooth projective curve C and obtained the well known Vafa-Intriligator formula. Our main computational technique will be virtual localization, a limiting procedure in intersection theory in which enumerative invariants are computed in terms of data from the fixed locus of a torus action. While such techniques would only yield results for toric surfaces such as P² and P¹ x P¹, we extend these results using cobordism classes to any surface S for which Quots admits a virtual fundamental class. In particular, our results hold over any Fano surface. The examples computed here outline a computational algorithm for further enumerative invariants on Quots

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