The Curve Exclusion Theorem for elliptic and K3 fibrations birational to Fano 3-fold hypersurfaces

TL;DR

This paper proves the Curve Exclusion Theorem, a key technical result essential for classifying elliptic and K3 fibrations birational to Fano 3-fold hypersurfaces, advancing the understanding of their birational geometry.

Contribution

It provides a complete proof of the previously unpublished Curve Exclusion Theorem, crucial for the classification of certain fibrations in algebraic geometry.

Findings

Complete proof of the Curve Exclusion Theorem

Applications to classification of elliptic and K3 fibrations

Examples illustrating the theorem's use

Abstract

The theorem referred to in the title is a technical result that is needed for the classification of elliptic and K3 fibrations birational to Fano 3-fold hypersurfaces in weighted projective space. We present a complete proof of the Curve Exclusion Theorem, which first appeared in the author's unpublished PhD thesis [Ry02] and has since been relied upon in solutions to many cases of the fibration classification problem. We give examples of these solutions and discuss them briefly.