Toric Rings Arising from Cyclic Polytopes

Abstract

Let d and n be positive integers with n ≥ d + 1 and 𝒫 ⊂ ℝ d an integral cyclic polytope of dimension d with n vertices, and let K[𝒫] = K[ℤ≥0𝒜𝒫] denote its associated semigroup K-algebra, where 𝒜𝒫 = {(1, α) ∈ ℝ d+1: α ∈ 𝒫} ∩ ℤ d+1 and K is a field. In the present paper, we consider the problem when K[𝒫] is Cohen–Macaulay by discussing Serre's condition (R 1), and we give a complete characterization when K[𝒫] is Gorenstein. Moreover, we study the normality of the other semigroup K-algebra K[Q] arising from an integral cyclic polytope, where Q is a semigroup generated by its vertices only.