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

Abstract

Tilting bundles provide a fundamental bridge between algebraic geometry and representation theory. For a tilting bundle on a smooth proper $d$-dimensional variety, the global dimension of its endomorphism algebra is at least $d$, and the most meaningful case is when this lower bound is attained. Such a tilting bundle, called a $d$-tilting bundle, fits into the framework of the derived McKay correspondence and higher Auslander--Reiten theory. The first main result of this paper shows that the existence of such a bundle forces the variety to be weak Fano: more precisely, if a smooth proper $d$-dimensional variety admits a $d$-tilting bundle, then its anti-canonical bundle is semiample and big. As a consequence, the endomorphism algebra of a $d$-tilting bundle is $d$-representation tame, so the geometry naturally produces higher-dimensional analogues of extended Dynkin quivers. Second, we prove a converse in dimension two: every weak del Pezzo surface over an algebraically closed field admits a $2$-tilting bundle. Together, these results give an affirmative answer to a conjecture posed by Daniel Chan for the variety case: a smooth projective surface admits a $2$-tilting bundle if and only if it is a weak del Pezzo surface. As an application, we construct non-commutative crepant resolutions (NCCRs) of anti-canonical cones over Du Val del Pezzo surfaces. Such an NCCR is obtained as the $3$-Calabi--Yau completion of the endomorphism algebra of a $2$-tilting bundle on the corresponding weak del Pezzo surface. This extends the known construction for smooth del Pezzo surfaces to the Du Val case and places Du Val del Pezzo cones within the framework of the derived McKay correspondence via higher Auslander--Reiten theory.