Tilting theory for hypersurface singularities of dimension one

It was shown by Buchweitz, the first author, and Yamaura that any N-graded commutative Gorenstein ring R of Krull dimension one with R₀ a field admits a standard silting object V in the stable category CM\, \!₀ZR. Moreover, they proved that the object V is tilting if and only if the a-invariant a is non-negative. In this article, under the additional assumption that R is a hypersurface singularity, we give an explicit description of the endomorphism algebra of V and prove that it is Iwanaga-Gorenstein of self-injective dimension at most $2$. In the case of where a is negative, we give an explicit description of the endomorphism dg algebra of V and prove that it is Gorenstein. Moreover, we give a characterization of Gorensteinness of homologically finite dg algebras in terms of Serre functors.