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Browsing Department of Mathematics by Subject "algebraic geometry"
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Item Open Access Abelian surfaces with real multiplication over finite fields(Colorado State University. Libraries, 2014) Freese, Hilary, author; Achter, Jeffrey, advisor; Pries, Rachel, committee member; Peterson, Chris, committee member; Tavani, Daniele, committee memberGiven a simple abelian surface A/Fq, the endomorphism algebra, End(A) ⊗ Q, contains a unique real quadratic subfield. We explore two different but related questions about when a particular real quadratic subfield K+ is the maximal real subfield of the endomorphism algebra. First, we compute the number of principally polarized abelian surfaces A/Fq such that K+ ⊂ End(A) ⊗ Q. Second, we consider an abelian surface A/Q, and its reduction Ap = A mod p, then ask for which primes p is K+ ⊂ End(A) ⊗ Q. The result from the first question leads to a heuristic for the second question, namely that the number of p < χ for which K+ ⊂ End(A) ⊗ Q grows like √χ/log(c).Item Open Access Computing syzygies of homogeneous polynomials using linear algebra(Colorado State University. Libraries, 2014) Hodges, Tim, author; Bates, Dan, advisor; Peterson, Chris, committee member; Böhm, A. P. Willem, committee memberGiven a ideal generated by polynomials ƒ1,...,ƒn in polynomial ring of m variables a syzygy is an n-tuple α1,.., αn, & αi in our polynomial ring of m variables such that our n-tuple holds the orthogonal property on the generators above. Syzygies can be computed by Buchberger's algorithm for computing Gröbner Bases. However, Gröbner bases have been computationally impractical as the number of variables and number of polynomials increase. The aim of this thesis is to describe a way to compute syzygies without the need for Grobner bases but still retrieve some of the same information as Gröbner bases. The approach is to use the monomial structure of the polynomials in our generating set to build syzygies using Nullspace computations.Item Open Access Finitely generated modules over Noetherian rings: interactions between algebra, geometry, and topology(Colorado State University. Libraries, 2020) Flores, Zachary J., author; Peterson, Christopher, advisor; Duflot, Jeanne, committee member; Cavalieri, Renzo, committee member; Ross, Kathryn, committee memberIn this dissertation, we aim to study finitely generated modules over several different Noetherian rings and from varying perspectives. This work is divided into four main parts: The first part is a study of algebraic K-theory for a certain class of local Noetherian rings; the second discusses extending well-known results on Lefschetz properties for graded complete intersection algebras to a class of graded finite length modules using geometric techniques; the third discusses the structure of various algebraic and geometric invariants attached to the finite length modules from the previous section; and lastly, we discuss the structure of annihilating ideals of classes of hyperplane arrangements in projective space.Item Open Access Preconditioning polynomial systems using Macaulay dual spaces(Colorado State University. Libraries, 2015) Ihde, Steven L., author; Bates, Daniel J., advisor; Peterson, Chris, committee member; Hulpke, Alexander, committee member; Young, Peter, committee memberPolynomial systems arise in many applications across a diverse landscape of subjects. Solving these systems has been an area of intense research for many years. Methods for solving these systems numerically fit into the field of numerical algebraic geometry. Many of these methods rely on an idea called homotopy continuation. This method is very effective for solving systems of polynomials in many variables. However, in the case of zero-dimensional systems, we may end up tracking many more solutions than actually exist, leading to excess computation. This project preconditions these systems in order to reduce computation. We present the background on homotopy continuation and numerical algebraic geometry as well as the theory of Macaulay dual spaces. We show how to turn an algebraic geometric preconditioning problem into one of numerical linear algebra. Algorithms for computing an H-basis and thereby preconditioning the original system to remove extraneous calculation are presented. The concept of the Closedness Subspace is introduced and used to replace a bottleneck computation. A novel algorithm employing this method is introduced and discussed.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 Toward a type B(n) geometric Littlewood-Richardson Rule(Colorado State University. Libraries, 2007) Davis, Diane E., author; Kley, Holger, advisorWe conjecture a geometric Littlewood-Richardson Rule for the maximal orthogonal Grassmannian and make significant advances in the proof of this conjecture. We consider Schubert calculus in the presence of a nondegenerate symmetric bilinear form on an odd-dimensional vector space (the type Bn setting) and use degenerations to understand intersections of Schubert varieties in the odd orthogonal Grassmannian. We describe the degenerations using combinatorial objects called checker games. This work is closely related to Vakil's Geometric Littlewood-Richardson Rule (Annals of Mathematics, 164).