Asymptotic regularity of graded families of ideals
Tai Huy Ha, Hop D. Nguyen, Thai Thanh Nguyen
TL;DR
The paper investigates the asymptotic behavior of Castelnuovo–Mumford regularity across graded families of homogeneous ideals in standard graded rings, proving existence results under Noetherian Rees algebras and various non-Noetherian conditions, and offering a Newton–Okounkov geometric perspective. It shows that when the Rees algebra is Noetherian (and Ann$(I_n)=0$) the limit $\lim_{n\to\infty} \mathrm{reg}(I_n)/n$ exists and matches asymptotic maximal generating degrees and integral-closure limits. For monomial and certain m-primary or Cohen–Macaulay families, the limit can be read off from the Newton–Okounkov region: $\lim_{n\to\infty} \mathrm{reg}(I_n)/n = \delta(\Delta(\mathcal{I}))$. The authors also present negative results: explicit counterexamples show nonexistence of limits for $\mathrm{reg}(Q^n+(f^n))/n$ and $\mathrm{reg}(Q^n\cap (f^n))/n$ in a characteristic-2 four-variable setting, and discuss implications for symbolic powers and Herzog–Hoa–Trung questions. Overall, the work connects asymptotic regularity to combinatorial convex geometry and Gröbner-basis techniques, yielding both broad existence results and delicate, explicit counterexamples.
Abstract
We show that the asymptotic regularity of a graded family $(I_n)_{n \ge 0}$ of homogeneous ideals in a reduced standard graded algebra, i.e., the limit $\lim_{n \rightarrow \infty} \text{reg } I_n/n$, exists in several cases; for example, when the family $(I_n)_{n \ge 0}$ consists of artinian ideals, or Cohen-Macaulay ideals of the same codimension over an uncountable base field of characteristic $0$, or when its Rees algebra is Noetherian. Many applications, including simplifications and generalizations of previously known results on symbolic powers and integral closures of powers of homogeneous ideals, are discussed. We provide a combinatorial interpretation of the limit $\lim_{n \rightarrow \infty} \text{reg } I_n/n$ in terms of the associated Newton--Okounkov region in various situations. We give a negative answer to the question of whether the limits $\lim_{n \rightarrow \infty} \text{reg } (I_1^n + \dots + I_p^n)/n$ and $\lim_{n \rightarrow \infty} \text{reg } (I_1^n \cap \cdots \cap I_p^n)/n$ exist, for $p \ge 2$ and homogeneous ideals $I_1, \dots, I_p$. We also examine ample evidence supporting a negative answer to the question of whether the asymptotic regularity of the family of symbolic powers of a homogeneous ideal always exists. Our work presents explicit Gröbner basis construction for ideals of the form $Q^n + (f^k)$, where $Q$ is a monomial ideal, $f$ is a polynomial in the polynomial ring in 4 variables over a field of characteristic $2$.