On the slope inequalities for extremal curves

TL;DR

This paper investigates whether extremal curves in projective space violate slope inequalities, concluding that in many cases they do not, based on geometric analysis and specific examples on Hirzebruch surfaces.

Contribution

It provides a detailed geometric analysis showing that extremal curves often do not violate slope inequalities, addressing a question posed by Kato and Martens.

Findings

Many extremal curves do not violate slope inequalities.

Certain curves on Hirzebruch surfaces satisfy the inequalities.

The analysis clarifies conditions under which violations occur or are absent.

Abstract

The present paper concerns the question of the violation of the r-th inequality for extremal curves in the projective r-space, posed by T. Kato and G. Martens. We show that the answer is negative in many cases. The result is obtained by a detailed analysis of the geometry of extremal curves and their canonical model. As a consequence, we show that particular curves on a Hirzebruch surface do not violate the slope inequalities in a certain range.