Sections and unirulings of families over $\mathbb{P}^1$

TL;DR

This paper investigates the geometric properties of families over the projective line, showing that under certain conditions on singular fibers, the total space is uniruled and admits sections or multisections, using symplectic cohomology techniques.

Contribution

It establishes new criteria linking the number of singular fibers in a family to the existence of sections or multisections, utilizing advanced symplectic cohomology methods.

Findings

If at most one singular fiber, then the total space is uniruled and admits sections.

If at most two singular fibers and certain Chern class conditions, then multisections exist.

Vanishing symplectic cohomology groups imply the existence of unirulings and sections.

Abstract

We consider morphisms $\pi: X \to \mathbb{P}^1$ of smooth projective varieties over $\mathbb{C}$. We show that if $\pi$ has at most one singular fibre, then $X$ is uniruled and $\pi$ admits sections. We reach the same conclusions, but with genus zero multisections instead of sections, if $\pi$ has at most two singular fibres, and the first Chern class of $X$ is supported in a single fibre of $\pi$. To achieve these result, we use action completed symplectic cohomology groups associated to compact subsets of convex symplectic domains. These groups are defined using Pardon's virtual fundamental chains package for Hamiltonian Floer cohomology. In the above setting, we show that the vanishing of these groups implies the existence of unirulings and (multi)sections.