Problems on pencils of small genus

Abstract

Let f : X → B be a fibre space of curves over a base curve B; write F = f−1(P ) for a fibre of f , and set g = genusF ≥ 2 and b = genusB. For genus 2 pencils, the geometric methods of Horikawa and Xiao Gang are very powerful and practical, and my aim is to try to generalise these to genus 3, 4, 5,. . . My problems here are concerned with the study of the relative canonical algebra R(X/B) = R(f) of f , and how to use it to get geographical information on X. First the definitions: write KX/B = KX −fKB, and set f∗(K X/B) = Rn for n ≥ 0.