# Tilting chains of negative curves on rational surfaces

## TL;DR

This paper introduces exact tilting objects on rational surfaces, constructed from chains of negative rational curves, establishing equivalences with module categories and advancing the understanding of derived categories in algebraic geometry.

## Contribution

It constructs exact tilting objects from chains of negative rational curves, linking geometric configurations to algebraic module categories and providing new tools for derived category analysis.

## Key findings

- Constructed exact tilting objects from chains of negative rational curves.
- Established equivalences with module categories over known algebras.
- Provided a new approach to study derived categories of rational surfaces.

## Abstract

We introduce the notion of exact tilting objects, which are partial tilting objects $T$ inducing an equivalence between the abelian category generated by $T$ and the category of modules over the endomorphism algebra of $T$. Given a chain of sufficiently negative rational curves on a rational surface, we construct an exceptional sequence whose universal extension is an exact tilting object. For a chain of (-2)-curves, we obtain an equivalence with modules over a well known algebra.