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

TL;DR

The paper characterizes when a smooth projective surface admits a $2$-tilting bundle, proving Cha’s conjecture that this occurs if and only if the surface is weak del Pezzo. It then constructs $2$-tilting bundles on all weak del Pezzo surfaces, using slope stability and iterated universal (co)extensions, and shows these tilting bundles induce tiltings on the total space of the canonical bundle. A key application is that the corresponding endomorphism algebras give $2$-representation infinite algebras whose $3$-Calabi-Yau completions yield NCCRs for Du Val del Pezzo cones, extending noncommutative resolution theory to singular cases. The work unifies higher representation theory with the geometry of del Pezzo surfaces and provides explicit NCCRs for a class of surface singularities through tilting and CY-completion techniques.

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.

Theorems & Definitions (43)

• Theorem 1
• Theorem 2
• Definition 1.1
• Definition 1.2
• Proposition 1.3
• Lemma 1.4
• proof
• proof : Proof of Proposition \ref{['SerreRI']}
• Proposition 1.5
• proof