Full exceptional collections on Fano threefolds and groups generated by spherical twists on K3 surfaces

TL;DR

This paper investigates the structure of full exceptional collections on certain Fano threefolds, showing they consist of shifted vector bundles, and analyzes the group generated by spherical twists on K3 surfaces.

Contribution

It proves that all full exceptional collections on these Fano threefolds are composed of shifted vector bundles and characterizes the group generated by spherical twists as free with explicit generators.

Findings

All full exceptional collections are shifted vector bundles.

The group generated by spherical twists is free.

Explicit generators for the spherical twist group are provided.

Abstract

For a Fano threefold admitting a full exceptional collection of vector bundles of length four we show that all full exceptional collections consist of shifted vector bundles. We prove this via a detailed study of the group generated by spherical twists on an anticanonical divisor. For example, we prove that this group is free and provide explicit generators.