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Let  S  be a minimal complex surface of general type with irregularity  q  ⩾ 2 and let  C  ⊂  S  be an irreducible curve of geometric genus  g . Assume that  C  is  Albanese defective , that is, 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   S   C  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 Lu [‘On surfaces of general type with maximal Albanese dimension’,  J. reine angew. Math.  641 (2010) 163–175.].

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