We propose to conduct research in the area of algebraic geometry; in particular the study of plane quartic curves predominantly over finite fields. The aim of this research is primarily to address Problems 1 and 2.Problem 1. How many rational points are there on a plane quartic curve over finite fields?Problem 2. The projective classification of plane quartic curves over finite fields.Details of Problems 1 and 2There are several bounds on the number of points on a plane quartic curve over a finite field of order q and for some instances of q there are sharpness results for these bounds. These bounds are given in Section 2 of the research proposal. There are no results about the existence of plane quartic curves with n rational points over a finite field of order q, where n is between these bounds, excluding sharpness results.The plane quartic curves are projectively classified over finite fields of small order; the smallest field over which the plane quartic curves are not projectively classified is of order eleven. There is not a classification of plane quartic curves over a general finite field as there is for plane cubic curves. Approaches to Problems 1 and 2The following gives an outline of the approaches to be taken in tackling Problems 1 and 2. As is made explicit in the research proposal, this is subject to change, as some approaches may prove fruitless and other approaches and techniques could arise during the research.The aim of studying the projective classification of plane quartic curves over fields of order smaller than eleven and determine if there exist any results that can be extrapolated to more general finite fields.The study of plane quartic curves over the complex numbers may reveal results that can be translated to plane quartic curves over finite fields.Are there any results to Problems 1 and 2 that apply over fields of specific order q; for example, q is a prime.Non-singular plane quartic curves possess a genus of 3, so non-singular plane quartic curves may be studied through the more general study of curves of genus 3.Lines in a cubic surfaces project onto bitangents of plane quartic curves; this is described in Section 3 of the research proposal together with a brief account of what is known about the number of lines on a cubic surface and the number of bitangents on the plane quartic curve onto which it projects.This leads to the question: Can the problem of determining how many rational points there are on a non-singular plane quartic curve or the classification of non-singular plane quartic curves be addressed through a study of their bitangents or through a study of the cubic surfaces.What can be determined about the classification of plane quartic curves over finite fields through a detailed study of the classification of plane cubic curves over finite fields?
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