Niveau: Supérieur, Doctorat, Bac+8
ASYMPTOTIC BEHAVIOUR OF RATIONAL CURVES DAVID BOURQUI Abstract. These are a preliminary version of notes for a course delivered during the summer school on rational curves held in 2010 at Institut Fourier, Grenoble. Any comments are welcomed. 1. Introduction 1.1. The problem. The problem we will be concerned with, which is also consid- ered in Peyre's lecture, may be loosely stated as follows: given an algebraic variety X (defined over a field k) possessing a lot of rational curves (by this we mean that the union of rational curves on X is not contained in a proper Zariski closed subset ; for example, this holds for rational varieties) is it possible to give a quantitative estimate of the number of rational curves on it? We expect of course an answer slighly less vague than: the number is infinite. To give a more precise meaning to the above question, fix a projective embedding ? : X ? Pn (or, if you prefer and which amounts almost to the same, an ample line bundle L on X). Then given a morphism x : P1 ? X we define its degree (with respect to ?) deg?(x) def = deg((x ? ?)?OPn(1) (1.1.1) (or degL(x) def = deg(x?L)). This is a nonnegative integer.
- zeta function
- dimension
- over
- original geometric
- geometric effective
- euler-poincaré characteristic
- function defined
- open dense
- positive constant