Gorenstein binomial edge ideals

Abstract

We classify connected graphs 𝐺 whose binomial edge ideal is Gorenstein. In our proofs we use Frobenius type techniques and F-pure thresholds. K E Y W O R D Sbinomial edge ideal, 𝐹-pure thresholds, gorenstein ideal, graded rings, initial ideals M S C ( 2 0 2 0 )

Gröbner deformations and F‐singularities

“…Let (𝑅, 𝔪) be an ℕ-graded ring with 𝑅 0 = 𝐾. Suppose that 𝑅∕𝑥𝑅 is 𝐹-full (strongly 𝐹-injective) for some homogeneous element 𝑥 of degree The conclusion that 𝑆∕𝐼 is 𝐹-injective provided in < (𝐼) is a squarefree monomial ideal has been proved independently in [19,Theorem 4.3], where this result has been crucial to prove that the only Gorenstein binomial edge ideals are complete intersections.…”

“…It is not clear to us how to show this fact (certainly known in the local case) without using Theorem 4.3. The conclusion that $$S/I$$ is F ‐injective provided $$\mathrm{in}_&lt;(I)$$ is a squarefree monomial ideal has been proved independently in [19, Theorem 4.3], where this result has been crucial to prove that the only Gorenstein binomial edge ideals are complete intersections.…”

Frobenius methods in combinatorics

“…For example, we discuss several properties of the ordinary and symbolic powers of determinantal ideals (see Section 5). We also present the characterization of Gorenstein binomial edge ideals obtained by González-Martínez [42] (see Theorem 3.7). We give a prime characteristic proof of a result by Sturmfels on the regularity of normal toric varieties.…”