Browsing by Author "Cavalieri, Renzo, advisor"
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Item Open Access A quantum H*(T)-module via quasimap invariants(Colorado State University. Libraries, 2024) Lee, Jae Hwang, author; Shoemaker, Mark, advisor; Cavalieri, Renzo, advisor; Gillespie, Maria, committee member; Peterson, Christopher, committee member; Hulpke, Alexander, committee member; Chen, Hua, committee memberFor X a smooth projective variety, the quantum cohomology ring QH*(X) is a deformation of the usual cohomology ring H*(X), where the product structure is modified to incorporate quantum corrections. These correction terms are defined using Gromov-Witten invariants. When X is toric with geometric quotient description V//T, the cohomology ring H*(V//T) also has the structure of a H*(T)-module. In this paper, we introduce a new deformation of the cohomology of X using quasimap invariants with a light point. This defines a quantum H*(T)-module structure on H*(X) through a modified version of the WDVV equations. We explicitly compute this structure for the Hirzebruch surface of type 2. We conjecture that this new quantum module structure is isomorphic to the natural module structure of the Batyrev ring for a semipositive toric variety.Item Open Access Bridgeland stability of line bundles on smooth projective surfaces(Colorado State University. Libraries, 2014) Miles, Eric W., author; Cavalieri, Renzo, advisor; Achter, Jeff, committee member; Peterson, Chris, committee member; Prasad, Ashok, committee member; Pries, Rachel, committee memberBridgeland Stability Conditions can be thought of as tools for creating and varying moduli spaces parameterizing objects in the derived category of a variety X. Line bundles on the variety are fundamental objects in its derived category, and we characterize the Bridgeland stability of line bundles on certain surfaces. Evidence is provided for an analogous characterization in the general case. We find stability conditions for P1 × P1 which can be seen as giving the stability of representations of quivers, and we deduce projective structure on the Bridgeland moduli spaces in this situation. Finally, we prove a number of results on objects and a construction related to the quivers mentioned above.Item Open Access Combinatorial structures of hyperelliptic Hodge integrals(Colorado State University. Libraries, 2021) Afandi, Adam, author; Cavalieri, Renzo, advisor; Shoemaker, Mark, advisor; Adams, Henry, committee member; Prasad, Ashok, committee memberThis dissertation explores the combinatorial structures that underlie hyperelliptic Hodge integrals. In order to compute hyperelliptic Hodge integrals, we use Atiyah-Bott (torus) localization on a stack of stable maps to [P1/Z2] = P1 × BZ2. The dissertation culminates in two results: a closed-form expression for hyperelliptic Hodge integrals with one λ-class insertion, and a structure theorem (polynomiality) for Hodge integrals with an arbitrary number of λ-class insertions.Item Open Access Graduate students' representational fluency in elliptic curves(Colorado State University. Libraries, 2023) Dawson, Erin, author; Cavalieri, Renzo, advisor; Ellis Hagman, Jessica, advisor; Zarestky, Jill, committee memberElliptic curves are an important concept in several areas of mathematics including number theory and algebraic geometry. Within these fields, three mathematical objects have each been referred to as an elliptic curve: a complex torus, a smooth projective curve of degree 3 in P2 with a chosen point, and a Riemann surface of genus 1 with a chosen point. In number theory and algebraic geometry, it can be beneficial to use different representations of an elliptic curve in different situations. This skill of being able to connect and translate between mathematical objects is called representational fluency. My work explores graduate students' representational fluency in elliptic curves and investigates the importance of representational fluency as a skill for graduate students. Through interviews with graduate students and experts in the field, I conclude 3 things. First, some of the connections between the above representations are made more easily by graduate students than other connections. Second, students studying number theory have higher representational fluency in elliptic curves. Third, there are numerous benefits of representational fluency for graduate students.Item Open Access Intersections of ψ classes on Hassett spaces of rational curves(Colorado State University. Libraries, 2018) Sharma, Nand, author; Cavalieri, Renzo, advisor; Peterson, Chris, committee member; Achter, Jeff, committee member; Prasad, Ashok, committee memberHassett spaces are moduli spaces of weighted stable pointed curves. In this work, we consider such spaces of curves of genus 0 with weights all 1/q , q being a positive integer greater than or equal to 2. These spaces are interesting as they have different universal families and different intersection theory when compared with classical moduli spaces of pointed stable rational curves. We develop closed formulas for intersections of ψ-classes on such spaces. In our main result, we encode the formula for top intersections in a generating function obtained by applying an exponential differential operator to the Witten-potential.Item Open Access Moduli spaces of rational graphically stable curves(Colorado State University. Libraries, 2021) Fry, Andy J., author; Cavalieri, Renzo, advisor; Shoemaker, Mark, committee member; Wilson, James, committee member; Tavani, Daniele, committee memberWe use a graph to define a new stability condition for the algebraic and tropical moduli spaces of rational curves. Tropically, we characterize when the moduli space has the structure of a balanced fan by proving a combinatorial bijection between graphically stable tropical curves and chains of flats of a graphic matroid. Algebraically, we characterize when the tropical compactification of the compact moduli space agrees with the theory of geometric tropicalization. Both characterization results occur only when the graph is complete multipartite.Item Open Access Open and closed Gromov-Witten theory of three-dimensional toric Calabi-Yau orbifolds(Colorado State University. Libraries, 2013) Ross, Dustin J., author; Cavalieri, Renzo, advisor; Achter, Jeff, committee member; Lunkenheimer, Erika, committee member; Peterson, Chris, committee memberWe develop the orbifold topological vertex, an algorithm for computing the all-genus, open and closed Gromov-Witten theory of three-dimensional toric Calabi-Yau orbifolds. We use this algorithm to study Ruan's crepant resolutions conjecture and the orbifold Gromov-Witten/Donaldson-Thomas correspondence.Item Open Access Properties of tautological classes and their intersections(Colorado State University. Libraries, 2019) Blankers, Vance T., author; Cavalieri, Renzo, advisor; Achter, Jeff, committee member; Pries, Rachel, committee member; Shoemaker, Mark, committee member; Heineman, Kristin, committee memberThe 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.Item Open Access Pseudostable Hodge classes(Colorado State University. Libraries, 2024) Williams, Matthew M., author; Cavalieri, Renzo, advisor; Gillespie, Maria, committee member; Peterson, Chris, committee member; Tucker, Dustin, committee memberWe study the relationship between Hodge classes on moduli spaces of pseudostable and stable curves given by the contraction morphism T. While Mumford's relation does not hold in the pseudostable case, we show that one can express the (pullback via T of the) Chern classes of E ⊕ E^∨ solely in terms of strata and ψ classes. We organize the combinatorial structure of the pullback of products of two pseudostable λ classes and obtain an explicit comparison of arbitrary pseudostable and stable quadratic Hodge integrals, as well as certain families of cubic and higher degree pseudostable Hodge integrals.