# Examples of smooth components of moduli spaces of stable sheaves

## TL;DR

This paper demonstrates that in four specific cases, the universal family of stable sheaves on a moduli space can be viewed as a flat family parametrized by the original variety, linking the geometry of the moduli space and the variety.

## Contribution

It provides explicit examples where the universal family induces a smooth component of a moduli space of stable sheaves on the original moduli space.

## Key findings

- Universal family E can be realized as a flat family on the moduli space
- X is identified with a smooth component of a moduli space of stable sheaves on M
- Examples illustrate the smoothness of certain components of moduli spaces

## Abstract

Let M be a projective fine moduli space of stable sheaves on a smooth projective variety X with a universal family E. We prove that in four examples, E can be realized as a complete flat family of stable sheaves on M parametrized by X, which identifies X with a smooth connected component of some moduli space of stable sheaves on M.