Weak del Pezzo surfaces yield 2-hereditary algebras and 3-Calabi-Yau algebras

Abstract: The importance of studying $d$-tilting bundles, which are tilting bundles whose endomorphism algebras have global dimension $d$ (or less), on $d$-dimensional smooth projective varieties has been recognized recently. In Chan's paper, it is conjectured that a smooth projective surface has a $2$-tilting bundle if and only if it is weak del Pezzo. In this paper, we prove this conjecture. Moreover, we show that this endomorphism algebra becomes a $2$-representation infinite algebra whose 3-Calabi-Yau completion gives a non-commutative crepant resolution (NCCR) of the corresponding Du Val del Pezzo cone.