Gorenstein \(T\)-spread Veronese algebras (Q2215993)

Let \(S=K[x_1,\dots,x_n]\) and \(t\geq 0\) be an integer. The notion of \(t\)-spread monomial was introduced in [\textit{V. Ene} et al., Commun. Algebra 47, No. 12, 5303--5316 (2019; Zbl 1426.13004)]. A monomial written as \(x_{i_1}x_{i_2}\cdots x_{i_d}\), with \(i_1\leq i_2\leq \cdots \leq i_d\), is \(t\)-spread if and only if \(i_j-i_{j-1}\geq t\), for all \(2\leq j\leq d\). (Accordingly, any monomial is \(0\)-spread and any square-free monomial is \(1\)-spread.) In the article under review, the author considers \(K[I_{n,d,t}]\), the graded sub-algebra of \(S\) generated by the set of all \(t\)-spread monomials of degree \(d\), which is called a \(t\)-spread Veronese algebra. The main result states that \(K[I_{n,d,t}]\), with \(t\geq 2\), is Gorenstein if and only if \[ n\in \{(d-1)t+1,(d-1)t+2,dt,dt+1,dt+d\}. \] (Theorem 3.4). If \(n=(d-1)t+1\), there is only one \(t\)-spread monomial: \[ x_1x_{t+1}x_{2t+1}\cdots x_{(d-1)t+1} \] and so \(K[I_{n,d,t}]\) is a polynomial ring in one variable. If \(n=(d-1)t+2\), then \(K[I_{n,d,t}]\) is also a polynomial ring (Corollary 2.2). By the same result no other case listed above yields a polynomial ring. For the proof of the main result, the author uses the theory of Ehrhart rings.