Properties of tautological classes and their intersections
Date
2019
Authors
Blankers, Vance T., author
Cavalieri, Renzo, advisor
Achter, Jeff, committee member
Pries, Rachel, committee member
Shoemaker, Mark, committee member
Heineman, Kristin, committee member
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Abstract
The tautological ring of the moduli space of curves is an object of interest to algebraic geometers in Gromov-Witten theory and enumerative geometry more broadly. The intersection theory of this ring has a highly combinatorial structure, and we develop and exploit this structure for several ends. First, in Chapter 2 we show that hyperelliptic loci are rigid and extremal in the cone of effective classes on the moduli space of curves in genus two, while establishing the skeleton for similar results in higher genus. In Chapter 3 we connect the intersection theory of three families of important tautological classes (Ψ-, ω-, and κ-classes) at both the cycle and numerical level. We also show Witten's conjecture holds for κ-classes and reformulate the Virasoro operators in terms of κ-classes, allowing us to effectively compute relations in the κ-class subring. Finally, in Chapter 4 we generalize the results of the previous chapter to weighted Ψ-classes on Hassett spaces.
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Subject
intersection theory
tautological ring
algebraic geometry
Witten's conjecture
moduli space of curves