Measures of Irrationality and Vector Bundles on Trees of Rational Curves

Abstract

This thesis is divided into three parts. In the first, we define the covering gonality and separable covering gonality of varieties over fields of positive characteristic, generalizing the definition given by Bastianelli-de Poi-Ein-Lazarsfeld-Ullery for complex varieties. We show that over an arbitrary field a smooth degree d dimension n hypersurface has separable covering gonality at least d-n and a very general such hypersurface has covering gonality at least (d-n+1)/2. In the second part, a chapter joint with Isabel Vogt, 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 geometric points in C with [k(P):k] ≤e. Developing techniques that make use of an auxiliary smooth surface containing the curve, we show that this invariant can take any value subject to constraints imposed by the gonality. Building on work of Debarre--Klassen, we show that this invariant is equal to the gonality for all sufficiently ample curves on a surface S with trivial irregularity. In the final part, we study vector bundles on trees of smooth rational curves. We determine explicit conditions under which a vector bundle on a tree of smooth rational curves can be expressed as the flat limit of some particular vector bundle on P^1.