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dc.contributor.advisorBjorn Poonen and Joe Harris.en_US
dc.contributor.authorVogt, Isabel Marley.en_US
dc.contributor.otherMassachusetts Institute of Technology. Department of Mathematics.en_US
dc.date.accessioned2019-09-16T22:33:59Z
dc.date.available2019-09-16T22:33:59Z
dc.date.copyright2019en_US
dc.date.issued2019en_US
dc.identifier.urihttps://hdl.handle.net/1721.1/122168
dc.descriptionThesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2019en_US
dc.descriptionCataloged from PDF version of thesis.en_US
dc.descriptionIncludes bibliographical references (pages 209-212).en_US
dc.description.abstractThis thesis consists of four parts, roughly progressing from results in algebraic geometry to results in number theory. The unifying figure throughout this thesis is an algebraic curve: we begin with curves over the complex numbers and their realizations in projective space, move then to curves over number fields, and finally end with arithmetic questions concerning Galois properties of torsion points on elliptic curves. The first part generalizes the familiar fact that there always exists a line through 2 points in projective space, but not through 3 points in general position. We consider the more general fundamental incidence question: when does there exists a degree d, genus g curve though n general points in P? We work in the case when the curve is of general moduli, and hence by the Brill-Noether theorem p(d, g, r) = g - (r + 1)(g - d + r) >/_ 0. In this range, a dimension count predicts the maximal possible n.en_US
dc.description.abstractUsing deformation theory to translate this problem into a stability-like condition on the normal bundle of a general such curve, when r = 3 and 4 we prove that this naive dimension count is correct in all but two cases. The work in P⁴ is joint with Eric Larson. In the second part, we investigate an arithmetic analogue of the gonality of a smooth projective curve C over a number field k: the minimal e such there are infinitely many points of degree bounded by e. We call this invariant the arithmetic degree of irrationality of the curve C over k. Such an integer is always bounded by the gonality of the curve, since the preimage of the infinite set of rational points on P¹ lie in the set of points of residue degree at most the gonality. By work of Faltings [Fal94], Harris-Silverman [HS91] and Abramovich-Harris [AH91], it is well-understood when this invariant is 1, 2, or 3; by work of Debarre-Fahlaoui [DF93] these criteria do not generalize to e at least 4.en_US
dc.description.abstractIn this chapter, we develop techniques to compute this invariant that make use of an auxiliary smooth surface containing the curve. Using this idea, we show that this invariant can take any value subject to constraints imposed by the gonality. We then use these techniques to generalize work of Debarre-Klassen [DK94] and show that this invariant is equal to the gonality for all sufficiently positive curves on a surface S with trivial irregularity (i.e., discrete Picard group). This chapter is joint work with Geoffrey Smith.en_US
dc.description.statementofresponsibilityby Isabel Marley Vogt.en_US
dc.format.extent212 pagesen_US
dc.language.isoengen_US
dc.publisherMassachusetts Institute of Technologyen_US
dc.rightsMIT theses are protected by copyright. They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission.en_US
dc.rights.urihttp://dspace.mit.edu/handle/1721.1/7582en_US
dc.subjectMathematics.en_US
dc.titleSome results in the arithmetic and geometry of curvesen_US
dc.typeThesisen_US
dc.description.degreePh. D.en_US
dc.contributor.departmentMassachusetts Institute of Technology. Department of Mathematicsen_US
dc.identifier.oclc1117775150en_US
dc.description.collectionPh.D. Massachusetts Institute of Technology, Department of Mathematicsen_US
dspace.imported2019-09-16T22:33:57Zen_US
mit.thesis.degreeDoctoralen_US
mit.thesis.departmentMathen_US


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