  The Konno invariant of a projective variety X is the minimum geometric genus
of the fiber of a rational pencil on X. It was computed by Konno for surfaces
in P^3, and in general can be viewed as a measure of the complexity of X. We
estimate Konno(X) for some natural classes of varieties, including sharp
asymptotics for polarized K3 surfaces. In an appendix, we give a quick proof of
a classical formula due to Deligne and Hoskin for the colength of an integrally
closed ideal on a surface.
