Some results in the arithmetic and geometry of curves

Abstract

This 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.

Using 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.

In 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.