The full exceptional collections of categorical resolutions of curves

Abstract

This paper gives a complete answer of the following question: which (singular, projective) curves have a categorical resolution of singularities which admits a full exceptional collection? We prove that such full exceptional collection exists if and only if the geometric genus of the curve equals to 0. Moreover we can also prove that a curve with geometric genus equal or greater than 1 cannot have a categorical resolution of singularities which has a tilting object. The proofs of both results are given by a careful study of the Grothendieck group and the Picard group of that curve.Comment: 10 pages, minor change

Scalar extensions of categorical resolutions of singularities

“…The "if" part could be obtained by an explicit construction of a categorical resolution. In fact the proof is exactly the same as that of [17] Proposition 4.1.…”

“…One of the main results in this paper is the following proposition. As an application we study the categorical resolution of projective curves X over a non-algebraically closed field k. Using the technique of scalar extension we obtain the following theorems which generalize the main results in [17].…”

“…We also have the generalization of [17] Theorem 4.10 to non-algebraically closed base field. Theorem 5.5.…”

A Lie theoretical construction of a Landau–Ginzburg model without projective mirrors

“…4.6] gives that G 0 (Y ) is not finitely generated. If Y ′ = P 1 and Y = P 1 , then [18,Prop. 4.1] gives that a categorical resolution (in the sense of [14]) of D b (Coh Y ) has a full exceptional collection, but the proof of [18,Prop.…”

Singular Curves and Quasi-hereditary Algebras

“…According to [20, Corollary 4'], we have: gl.dim(Λ) ≤ gl.dim(Q) + 2.Remark 7.5. In a recent work[24, Theorem 4.10], the following inversion of Theorem 7.4 was obtained. Assume X is a projective curve over an algebraically closed field and Λ a finite dimensional -algebra of finite global dimension such that there exist functors…”