Dicritical divisors and hypercurvettes

TL;DR

This paper generalizes the classification of dicritical divisors and the construction of rational functions with prescribed dicritical components from surfaces to higher-dimensional varieties, using the concept of hypercurvettes.

Contribution

It extends the concept of curvettes to higher dimensions and proves a generalization of the classification of dicritical divisors for arbitrary dimension.

Findings

Generalization of dicritical divisor classification to higher dimensions.

Construction of rational functions with prescribed dicritical components.

Use of hypercurvettes in higher-dimensional settings.

Abstract

Germs of rational functions~$h$ on points $p$ of smooth varieties~$S$ define germs of rational maps to the projective line. Assume that $p$ is in the indeterminacy locus of $h$. If $\pi:\hat{S}\to S$ is a birational map which is an isomorphism outside $p$, then $h$ lifts to a germ of a rational map on $(\hat{S}, \pi^{-1}(p))$. The exceptional components $E_i$ of $\pi^{-1}(p)$ are classified according to the restriction of (the lift of) $h$ to $E_i$; the dicritical components are those where this restriction induces a dominant map. In a series of papers, Abhyankar and the first named author studied this setting in dimension $2$, where the main result is that, for any given $\pi$, there is a rational function $h$ with a prescribed subset of exceptional components that are dicritical of some given degree. The concept of curvette of an exceptional component played a key role in the proof. The second named author extended previously the concept of curvette to the higher dimensional case. Here we use this concept to generalize the above result to arbitrary dimension.