A uniform bound on the canonical degree of Albanese defective curves on surfaces

TL;DR

This paper establishes a linear upper bound on the canonical degree of Albanese defective curves on minimal complex surfaces of general type, relating it to the surface's invariants and the curve's genus, generalizing previous results.

Contribution

It provides a new uniform bound on the canonical degree of Albanese defective curves, extending and sharpening earlier bounds for curves with small genus.

Findings

Derived a linear upper bound in terms of K^2_S and g for the canonical degree K_SC.

Established bounds for curves with genus g <= q-1.

Generalized previous results to broader classes of curves.

Abstract

Let S be a minimal complex surface of general type with irregularity q>=2 and let C be an irreducible curve of geometric genus g contained in S. Assume that C is "Albanese defective", i.e., that the image of C via the Albanese map does not generate the Albanese variety Alb(S); we obtain a linear upper bound in terms of K^2_S and g for the canonical degree K_SC of C. As a corollary, we obtain a bound for the canonical degree of curves with g<= q-1, thereby generalizing and sharpening the main result of [S.Y. Lu, On surfaces of general type with maximal Albanese dimension, J. Reine Angew. Math. 641 (2010), 163-175].