Weak del Pezzo surfaces are characterized by the existence of $2$-tilting bundles

TL;DR

This paper characterizes weak del Pezzo surfaces via the existence of 2-tilting bundles, linking geometric properties to algebraic structures and providing non-commutative resolutions.

Contribution

It proves that a smooth projective surface admits a 2-tilting bundle if and only if it is a weak del Pezzo surface, confirming a conjecture by Daniel Chan.

Findings

A smooth projective surface admits a 2-tilting bundle iff it is a weak del Pezzo surface.

Weak Fano varieties admit a d-tilting bundle, generalizing the main result.

The cone of any Du Val del Pezzo surface admits a non-commutative crepant resolution.

Abstract

On $d$-dimensional smooth proper varieties, $d$-tilting bundles are important since they provide a bridge from the geometry of such varieties to the derived McKay correspondence and to higher Auslander-Reiten theory. Here, a $d$-tilting bundle means a tilting bundle whose endomorphism algebra has global dimension at most $d$. The main result of this paper gives an affirmative answer to a conjecture posed by Daniel Chan for variety case: a smooth projective surface admits a $2$-tilting bundle if and only if it is a weak del Pezzo surface. Moreover, we strengthen one direction: if a $d$-dimensional smooth proper variety admits a $d$-tilting bundle, then it is weak Fano. As an application, we show that the cone of any Du Val del Pezzo surface admits a non-commutative crepant resolution (NCCR). We obtain such an NCCR as the $3$-Calabi-Yau completion of the endomorphism algebra of a $2$-tilting bundle on the corresponding weak del Pezzo surface. This endomorphism algebra belongs to the class of $2$-representation tame algebras, which is a $2$-dimensional generalization of the path algebras of the extended Dynkin quivers.