The Fourier-Mukai transform of a universal family of stable vector bundles

TL;DR

This paper investigates the Fourier-Mukai transform associated with a universal family of stable vector bundles on a specific moduli space, demonstrating that it is not fully faithful, which impacts the understanding of derived equivalences.

Contribution

It provides a counterexample showing that the Fourier-Mukai transform from a universal family on a particular moduli space is not fully faithful, challenging assumptions about such transforms.

Findings

The Fourier-Mukai transform $\

It is not fully faithful for the given moduli space.

Implications for derived category equivalences and moduli space analysis.

Abstract

In this note we prove that the Fourier-Mukai transform $\Phi_{\mathcal{U}}$ induced by the universal family of the moduli space $\mathcal{M}_{\mathbb{P}^2}(4,1,3)$ is not fully faithful.