Demailly-Semple jets of orders 4 and 5 in dimension 2

TL;DR

This paper investigates Demailly-Semple jet bundles of orders 4 and 5 in two-dimensional complex manifolds to assess their algebraic properties related to Kobayashi's hyperbolicity conjecture, through detailed manual calculations.

Contribution

It provides new insights into the algebraic structure of high-order jets in dimension two, specifically addressing orders 4 and 5, which were previously less understood.

Findings

Some expectations about algebraic tameness are disproved.

Systematic calculations reveal complex algebraic behaviors.

Results contribute to understanding hyperbolicity in complex geometry.

Abstract

In view of Kobayashi's hyperbolicity conjecture, Demailly-Semple jets of orders 4 and 5 in dimension 2 are studied, some expectations about their algebraic tameness being (dis)proved, after systematic, substantial, formal, manual calculations.