the rank-k numerical range has a close connection to the construction of quantum error correction code for a noisy quantum channel. for noisy quantum channel, a quantum error correcting code of dimension k exists if and only if the associated joint rank-k numerical range is non-empty. in this paper the notion of joint rank-k numerical range is generalized and some statements of [2011, generaliz...

Using superisolated singularities we present examples and counterexamples to some of the most important conjectures regarding invariants of normal surface singularities. More precisely, we show that the " Seiberg-Witten invariant conjecture " (of Nicolaescu and the third author), the " Universal abelian cover conjecture " (of Neumann and Wahl) and the " Geometric genus conjecture " fail (at lea...

Let I be an m-primary ideal of a Noetherian local ring (R,m). We consider the Gorenstein and complete intersection properties of the associated graded ring G(I) and the fiber cone F (I) of I as reflected in their defining ideals as homomorphic images of polynomial rings over R/I and R/m respectively. In case all the higher conormal modules of I are free over R/I , we observe that: (i) G(I) is C...

We introduce the notion of relative singularity category with respect to any self-orthogonal subcategory ω of an abelian category. We introduce the Frobenius category of ω-Cohen-Macaulay objects, and under some reasonable conditions, we show that the stable category of ω-Cohen-Macaulay objects is triangle-equivalent to the relative singularity category. As applications, we relate the stable cat...

This paper is devoted to graded algebras A having a single homogeneous relation. We give a criterion for A to be N-Koszul where N is the degree of the relation. This criterion uses a theorem due to Gerasimov. As a consequence of the criterion, some new examples of N-Koszul algebras are presented. We give an alternative proof of Gerasimov’s theorem for N = 2, which is related to Dubois-Violette’...

We modify the definition of the families of A and B stringy cohomology spaces associated to a pair of dual reflexive Gorenstein cones. The new spaces have the same dimension as the ones defined in the joint paper with Mavlyutov [BoM], but they admit natural flat connections with respect to the appropriate parameters. This solves a longstanding question of relating GKZ hypergeometric system to s...

We discuss analogs of Faber’s conjecture for two nested sequences of partial compactifications of the moduli space of smooth curves. We show that their tautological rings are one-dimensional in top degree but do not satisfy Poincaré duality. The structure of the tautological ring of the moduli space of stable curves is predicted by the Faber conjecture, which states that R(Mg,n) is Gorenstein w...

Cohomology rings of various classes of groups have curious duality properties expressed in terms of their local cohomology [2, 3, 12, 4, 5, 6]. We formulate a purely algebraic form of this duality, and investigate its consequences. It is obvious that a Cohen-Macaulay ring of this sort is automatically Gorenstein, and that its Hilbert series therefore satisfies a functional equation, and our mai...