Terminal Quotient Singularities in Dimensions Three and Four

Abstract

We classify isolated terminal cyclic quotient singularities in dimension three, and isolated Gorenstein terminal cyclic quotient singularities in dimension four. In addition, we give a new proof of a combinatorial lemma of G. K. White using Bernoulli functions.

N = 3 $$ \mathcal{N}=3 $$ four dimensional field theories

“…Close to any of these fixed points we have a C 4 /Z 2 geometry, with the Z 2 inverting all coordinates of the C 4 . Notice that the C 4 /Z 2 singularity is terminal [13,14]: it admits no supersymmetric resolution or deformation into a smooth fourfold. This is in good agreement with the fact that there are no twisted sectors that could smooth out the O3 plane.…”

Canonical quotient singularities in dimension three

“…On the other hand, by the theorem of Khinich [5] and Watanabe [12], X/G is Gorenstein if and only if det g = 1 (which holds if and only if (ak/N) + (bk/N) + (ck/N) g Z for all k). Since we proved in [7] that X/G is terminal if and only if case (ii) in Theorem 3 holds, this theorem now follows immediately from Theorem 3. Q.E.D.…”

“…The combinatorial fact which we used in [7] to classify terminal quotient singularities is Theorem 2 [13, 3, 4 and 7]. Let a, b, c be integrers relatively prime to N. Suppose, for all k G Z -NZ, We can now state our main combinatorial result.…”

Untitled

“…Lattice-free polytopes are of importance in geometry of numbers and integer linear optimizationsee [Averkov et al 2011;Nill and Ziegler 2011] for recent results. Lattice-free simplices turn up naturally in singularity theory [Morrison and Stevens 1984]. Most prominently, the famous flatness theorem states that n-dimensional lattice-free convex bodies have bounded lattice width (we refer to [Barvinok 2002] for details).…”