The Konno invariant of some algebraic varieties

TL;DR

This paper studies the Konno invariant, a measure of the complexity of algebraic varieties, providing estimates for various classes including polarized K3 surfaces, and offers a new proof of a classical formula in algebraic geometry.

Contribution

It extends the computation of the Konno invariant to new classes of varieties and provides asymptotic estimates, especially for polarized K3 surfaces, along with a simplified proof of a classical formula.

Findings

Estimated Konno invariants for certain algebraic varieties

Derived sharp asymptotics for polarized K3 surfaces

Provided a quick proof of a classical colength formula

Abstract

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.