Tilting objects in graded singularity categories of certain toric Gorenstein singularities

We study a class of Gorenstein isolated singularities which are the quotients of generic and unimodular representations of the one-dimensional torus, or of the product of the one-dimensional torus with a finite abelian group. Based on the works of Špenko and van den Bergh [Invent. Math. 210 (2017), 3–67] and Mori and Ueyama [Adv. Math. 297 (2016), 54–92], we show that the graded singularity categories of these varieties admit tilting objects, and hence are triangulated equivalent to the perfect categories of some finite-dimensional algebras.

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