The degree of the generators of the canonical ring of surfaces of general type with p g = 0

Abstract

Upper bounds for the degree of the generators of the canonical rings of surfaces of general type were found by Ciliberto [7]. In particular it was established that the canonical ring of a minimal surface of general type with p g = 0 is generated by its elements of degree lesser or equal to 6, ([7];, Th. (3.6)). This was the best bound possible to obtain at the time, since Reider's results, [11], were not yet available. In this note, this bound is improved in some cases (Theorems (3.1), (3.2)). ¶ In particular it is shown that if K 2≥ 5, or if K 2≥ 2 and |2 K S | is base point free this bound can be lowered to 4. This result is proved by showing first that, under the same hypothesis, the degree of the bicanonical map is lesser or equal to 4 if K 2≥ 3, (Theorem (2.1)), implying that the hyperplane sections of the bicanonical image have not arithmetic genus 0. The result on the generation of the canonical ring then follows by the techniques utilized in [7].