On the slope inequalities for extremal curves

Abstract

The present paper concerns the question of the violation of the r-th inequality for extremal curves in mathbb {P}^r, posed in [KM]. We show that the answer is negative in many cases (Theorem 4.13 and Corollary 4.14). 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 (Theorem 6.4).