Duality for asymptotic invariants of graded families

Technical Summary: Duality for Asymptotic Invariants of Graded Families

Problem Statement
The paper investigates the duality between sequences of natural numbers arising in commutative algebra and algebraic geometry, specifically focusing on graded families of ideals. The central problem is to understand the relationship between subadditive sequences (such as initial degrees of symbolic powers) and superadditive sequences (such as Castelnuovo-Mumford regularity values or jet separation indices). While these sequences often appear in related contexts, their asymptotic growth factors—such as the Waldschmidt constant and asymptotic regularity—have historically been studied somewhat independently. The authors aim to establish a rigorous duality framework that interchanges these sequence types, inverts their asymptotic growth constants, and provides homological reformulations for long-standing conjectures regarding point configurations in projective space.

Methodology
The authors employ a multi-faceted approach combining combinatorial sequence theory, inverse system theory, and geometric invariant theory:

1. Sequence Duality Framework: The foundation is laid in Section 2, where the authors generalize transformations on sequences of natural numbers. For a sequence $\alpha = \{\alpha_n\}$, they define dual sequences $\overleftarrow{\alpha}_n = \inf\{d \mid \alpha_d \geq n\}$ and $\overrightarrow{\alpha}_n = \sup\{d \mid \alpha_d \leq n\}$. They prove that these transformations are mutual inverses and, crucially, that they interchange the classes of subadditive and superadditive sequences. Under mild hypotheses, these transformations reciprocate the asymptotic growth factors ($\lim \alpha_n/n$).

2. Differentially Closed Filtrations: In Section 4, the authors introduce the concept of differentially closed graded filtrations of ideals. A filtration $\mathcal{I} = \{I_n\}$ is differentially closed if applying differential operators of order $k$ to elements in $I_n$ yields elements in $I_{n-k}$. This class includes ordinary powers, symbolic powers (via the Zariski-Nagata theorem), differential powers, and integral Frobenius powers.

3. Inverse Systems and Macaulay-Matlis Duality: The authors utilize the theory of inverse systems (Macaulay duality). For a differentially closed filtration $\mathcal{I}$, they construct a sequence of ideals $L_s(\mathcal{I})$ in the dual ring (either the divided power algebra $D$ or the polynomial ring $S$ in characteristic zero). They prove that $L_s(\mathcal{I})$ is an ideal if and only if $\mathcal{I}$ is differentially closed. This construction links the initial degrees of the filtration to the regularity of the quotient by the inverse system ideals.

4. Geometric Applications: The methodology is applied to specific geometric settings, including the jet separation sequence for sets of points and the Seshadri constant. The authors relate the regularity of symbolic powers to the jet separation indices via the duality transformations established in Section 2.

Key Contributions and Results

• Generalized Sequence Duality (Theorem 2.5 & 2.6): The paper establishes that the transformations $\overleftarrow{\cdot}$ and $\overrightarrow{\cdot}$ map subadditive sequences to superadditive sequences (and vice versa) and invert their asymptotic growth factors. Specifically, if $\alpha$ is subadditive with growth factor $\hat{\alpha}$, then $\overrightarrow{\alpha}$ is superadditive with growth factor $\hat{\alpha}^{-1}$.

• Duality for Differentially Closed Filtrations (Theorem 4.9 & 4.20): The authors prove that for a differentially closed graded filtration $\mathcal{I} = \{I_n\}$, the sequence of initial degrees $\alpha_n = \alpha(I_n)$ is subadditive, while the sequence of end-degrees of the inverse system quotients, $\beta_s = \text{end}(D/L_s(\mathcal{I}))$, is superadditive. These sequences are dual to each other (shifted by $n$), and their asymptotic growth factors satisfy the reciprocal relation:
  $$\hat{\beta} = \frac{\hat{\alpha}}{\hat{\alpha} - 1} \quad \text{and} \quad \hat{\alpha} = \frac{\hat{\beta}}{\hat{\beta} - 1}$$
  This generalizes previous results by Emsalem and Iarrobino regarding symbolic powers of radical ideals.

• Reciprocity of Seshadri Constants and Asymptotic Regularity (Theorem 5.5, 5.7, Corollary 5.8): For a finite set of points $X \subset \mathbb{P}^N$, the paper establishes a duality between the sequence of jet separation indices $s(X, d)$ and the sequence of Castelnuovo-Mumford regularities of symbolic powers $\text{reg}(I(X)^{(k)})$. Consequently, the multipoint Seshadri constant $\varepsilon(X)$ is shown to be the reciprocal of the asymptotic regularity:
  $$\varepsilon(X) = \lim_{d \to \infty} \frac{s(X, d)}{d} = \left( \lim_{k \to \infty} \frac{\text{reg}(I(X)^{(k)})}{k} \right)^{-1}$$
  This provides a structural explanation for a known reciprocity relation.

• Homological Reformulations of Nagata–Iarrobino Conjectures (Section 6): The authors reformulate the Nagata conjecture (regarding the Waldschmidt constant of very general points in $\mathbb{P}^2$) and the Iarrobino conjecture (for higher dimensions) in terms of inverse systems. Specifically, they show that the conjectures are equivalent to the statement that the asymptotic regularity of the inverse system ideals $L_s(X)$ grows at a rate of $\frac{\sqrt[N]{r}}{\sqrt[N]{r} - 1}$. They also propose a homological reformulation (Conjecture 6.2) stating that for very general points, the Waldschmidt constant equals the asymptotic regularity ($\hat{\alpha} = \widehat{\text{reg}}$).

Significance and Claims
The paper claims to provide a unifying principle for various asymptotic invariants in algebraic geometry. By framing the relationship between initial degrees and regularity as a duality of sequences, the authors offer a new perspective on the containment problem between ordinary and symbolic powers, the resurgence of ideals, and the geometry of point configurations.

The significance lies in:

1. Unification: Connecting the Waldschmidt constant, asymptotic resurgence, Seshadri constants, and asymptotic regularity through a single duality mechanism.
2. Generalization: Extending the classical Emsalem-Iarrobino results on inverse systems of symbolic powers to the broader class of differentially closed filtrations.
3. Reformulation: Offering homological and inverse-system-based reformulations of the Nagata and Iarrobino conjectures, potentially opening new avenues for attack via the study of the ideals $L_s(X)$.

The authors explicitly note that while they recover known reciprocity relations (e.g., between Seshadri constants and asymptotic regularity), their contribution is identifying the underlying sequence duality that necessitates this relationship. They conclude by posing several open questions regarding the subadditivity of regularity sequences for general radical ideals and the geometric interpretation of generators for $L_s(\mathcal{I})$ in broader contexts.