Halphen Pencils on Weighted Fano Threefold Hypersurfaces

TL;DR

This paper classifies all pencils on general weighted hypersurfaces in projective space with degree equal to the sum of weights, focusing on those whose general members are surfaces of Kodaira dimension zero.

Contribution

It provides a complete classification of such pencils on weighted Fano threefold hypersurfaces, a previously unexplored area.

Findings

Identifies all pencils with Kodaira dimension zero surfaces

Classifies the structure of these pencils on weighted hypersurfaces

Enhances understanding of the geometry of weighted Fano threefolds

Abstract

We classify all pencils on a general weighted hypersurface in $\mathbb{P}(1,a_{1},a_{2},a_{3},a_{4})$ of degree $\sum_{i=1}^{4}a_{i}$ whose general members are surfaces of Kodaira dimension zero.