Bott vanishing for elliptic surfaces

TL;DR

This paper investigates how the presence of specific singular fibers in elliptic surfaces over P^1 influences Bott Vanishing, providing criteria based on geometric properties and ample line bundles.

Contribution

It establishes new criteria for Bott Vanishing on elliptic surfaces related to the types of singular fibers and geometric configurations.

Findings

Singular fibers like cuspidal fibers impact Bott Vanishing.

Geometric properties determine Bott Vanishing for ample line bundles.

Criteria are provided for surfaces with large self-intersection line bundles.

Abstract

We explore Bott Vanishing for elliptic surfaces over $\mathbb{P}^1$. We show that Bott Vanishing is singnificantly affected by the geometric properties that whether there exists certain type of singular fibers on the elliptic fibration such as cuspidal fibers. For ample line bundle on the surface with large self-intersection, these geometric properties give criteria for Bott Vanishing.