Quasi-Regular Valuations

• Quasi-regular valuations are rank-one discrete valuations characterized by a value group of the form δ·ℤ₊ and a finitely generated, toric Rees algebra.
• They unify algebraic, topological, and measure-theoretic regularity conditions to enable optimal constructions like algebraic tangent cones and slope-stability analyses.
• Applications span algebraic geometry and domain theory, facilitating precise degenerations, Harder–Narasimhan filtrations, and computable treatments of probabilistic measures.

A quasi-regular valuation is a valuation-theoretic construct that arises in several advanced contexts within algebraic geometry, commutative algebra, and domain theory. The terminology captures distinct, but related, notions in the study of algebraic varieties (where it generalizes divisorial and monomial valuations) and in the domain-theoretic treatment of spaces and probabilistic measures over quasi-Polish spaces. The unifying theme is a combination of topological, algebraic, and measure-theoretic regularity conditions ensuring both practical structure (such as finiteness or discrete gradings) and strong uniqueness properties for associated degenerations or measures.

1. Algebraic Structure of Quasi-Regular Valuations

Let $R$ be a finitely generated integral domain over an algebraically closed field $k$, with $X = \mathrm{Spec}~R$ and $x \in X$ a $k$-point. A valuation $v \colon R \to \Gamma \cup \{\infty\}$, with $\Gamma \subset \mathbb{R}_{\geq 0}$, is finitely generated if the semigroup $\Gamma = v(R \setminus \{0\})$ generates a lattice $M \simeq \mathbb{Z}^r$ in $\mathbb{R}$.

A quasi-regular valuation of rank $1$ is defined as a valuation $v$ for which $\Gamma = \delta \cdot \mathbb{Z}_{\geq 0}$ for some $\delta > 0$ (i.e., the value group is a discrete subgroup of $\mathbb{R}$ of rank $1$) (Hada, 1 Feb 2026). This is equivalent to demanding that $\delta := \min\{v(f) - v(g) > 0\}$ is positive and $$v(R \setminus \{0\}) = \delta \cdot \mathbb{Z}_{\geq 0}$$.

The Rees algebra attached to $v$ is

$$\mathrm{Rees}_v(R) = \bigoplus_{k\in\mathbb{Z}} a_{k\delta}(v) \cdot t^{-k\delta}$$

with $a_\lambda(v) = \{f \in R ~|~ v(f) \geq \lambda\}$. The central fibre is $\mathcal{X}_0 = \mathrm{Spec}\,\mathrm{gr}_v(R)$ where

$$\mathrm{gr}_v(R) = \bigoplus_{k\geq 0} a_{k\delta}/a_{(k+1)\delta}.$$

The associated Rees algebra is always finitely generated and toric, reflecting the underlying lattice structure (Hada, 1 Feb 2026).

2. Slope Stability and Harder–Narasimhan Filtration for Graded Modules

Within the context of quasi-regular valuations, a slope-stability framework is developed for finitely generated $\mathbb{R}$-graded modules $M = \bigoplus_{\lambda \in \mathbb{R}} M_\lambda$ over $\mathbb{R}_{\geq 0}$-graded algebras $A$. The key invariants are:

• $$\deg M := a_{n-1}(M) - \mathrm{rk}(M) \, a_{n-1}(A)$$,
• $\mu(M) := \deg M / a_n(M)$, where $a_j(M)$ are the coefficients in the asymptotic Riemann–Roch expansion of $\dim_k(M_{\leq x})$.

Each such module $M$ admits a unique Harder–Narasimhan filtration:

$$0 = M_0 \subset M_1 \subset \cdots \subset M_\ell = M$$

with associated graded pieces that are torsion-free and semistable, and the slopes strictly decreasing: $\mu(Q_1) > \cdots > \mu(Q_\ell)$ for $Q_i = M_i / M_{i-1}$ (Hada, 1 Feb 2026).

3. Existence and Uniqueness of Optimal Algebraic Tangent Cones

For a quasi-regular valuation $v$ on $R$ with index $\delta$, and a torsion-free $R$-module $M$, a geometric $v$-valuative function $$v_M \colon M \to \delta \cdot \mathbb{Z} \cup \{\infty\}$$ satisfies ultrametricity and $R$-linearity, with $\mathrm{gr}_{v_M}(M)$ finitely generated over $\mathrm{gr}_v(R)$. The key invariant is $\Phi(v_M) = \mu_{\max} - \mu_{\min}$, the difference between the maximal and minimal slopes of the graded pieces in the HN filtration.

The existence theorem asserts there exists $v_M$ such that $0 \leq \Phi(v_M) < \delta$. Achieving this uses successive Hecke transforms along the maximal-slope HN submodule, with each transform reducing $\Phi$ by at least $\delta$, ensuring termination as $\Phi$ takes values in a discrete subset of $\mathbb{R}$ (Hada, 1 Feb 2026).

Uniqueness is up to rigid twist: two such optimal $v$-valuative functions $v_M$ and $v_M'$ with $0 \leq \Phi(v_M), \Phi(v_M') < \delta$ differ by a constant shift $c \in \delta \mathbb{Z}$, or by a single Hecke transform and shift, depending on $\Phi(v_M) + \Phi(v_M')$ relative to $\delta$. Thus, the isomorphism class of the associated graded module (as a direct sum of stable reflexive pieces, up to grading shifts) is canonically attached to $(\mathcal{E}, v)$ for any torsion-free sheaf $\mathcal{E}$ (Hada, 1 Feb 2026).

4. Quasi-Regular Valuations in the Domain-Theoretic Framework

In the context of domain theory and quasi-Polish spaces, a valuation $\nu \colon \mathcal{O}(X) \to [0,\infty]$ on a topological space $X$ satisfies strictness, modularity, and Scott-continuity:

• $\nu(\varnothing) = 0$,
• $\nu(U) + \nu(V) = \nu(U \cup V) + \nu(U \cap V)$ for $U,V$ open,
• $\nu(\bigcup_i U_i) = \sup_i \nu(U_i)$ for any directed family $\{U_i\}$ (Brecht, 2021).

For sober, countably based (notably quasi-Polish) spaces, every Scott-continuous valuation is already "quasi-regular" in the sense that it is determined by its restriction to compact saturated sets. The space of such valuations $\mathcal{V}(X)$ is itself quasi-Polish and can be presented as an ideal space $\mathcal{I}_{\prec_V}$ for a computably defined transitive relation $\prec_V$ induced from a presentation of $X$ as an ideal space $\mathcal{I}_\prec$ (Brecht, 2021). This construction internalizes the probabilistic powerdomain viewpoint, enabling constructive and computable analysis of quasi-regular valuations.

5. Connections to Previous Frameworks and Examples

Quasi-regular valuations generalize the notion of the blow-up valuation studied by Chen–Sun, where $v_E$ is associated to the exceptional divisor $E$ of the blow-up at a smooth point, and $\Gamma = \mathbb{Z}_{\geq 0}$, $\delta = 1$. The algebraic tangent cone and optimal extension results for such divisorial valuations extend fully to any rank-one finitely generated valuation, including monomial valuations, valuations centered on singularities, and those induced by $\mathbb{C}^*$-actions on affine cones (Hada, 1 Feb 2026).

Table 1: Examples of Quasi-Regular Valuations

Example	$v$ (valuation) description	$\delta$ (index)
Monomial valuation on $A^2$	$v(x)=1$, $v(y)=2$, $\Gamma=\mathbb{Z}$	$1$
Weighted cone over projective variety	$v$ from $\mathbb{C}^*$-weight grading	$1$
Blow-up at a singular point	$v$ from exceptional divisor of resolution	$1$

6. Applications and Formalization Aspects

In algebraic geometry, quasi-regular valuations enable the construction of optimal $G_m$-equivariant degenerations of torsion-free sheaves, with canonical algebraic tangent cones determined up to grading shifts on stable summands. This is crucial for the study of degenerations, compactifications, and slope-stability in moduli theory (Hada, 1 Feb 2026).

In the domain-theoretic/quasi-Polish context, the ideal-space presentation of valuations allows for fully constructive, combinatorial, and even arithmetically formalizable treatments of measure and probabilistic powerdomain theory. The explicitness of the construction ensures compatibility with computability notions, supports the extension to computable measures, and provides deep connections with continuous lattices and monadic semantics (Brecht, 2021).

7. Classification Criteria and Structural Insights

A valuation on $R$ is quasi-regular exactly when it has rational rank $1$ and its value group is a discrete subgroup of $\mathbb{R}$, equivalently when $\mathrm{Rees}_v(R)$ is generated in a single grading variable $t^\delta$. Such valuations are prevalent in geometric and singularity-theoretic contexts, providing a unified framework for the precise construction of tangent cones and optimal degenerations. The topological and domain-theoretic generalization ensures that in quasi-Polish spaces, all Scott-continuous, modular valuations are automatically “quasi-regular” under this broader usage of the term (Hada, 1 Feb 2026, Brecht, 2021).