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arxiv: 2209.06044 · v3 · pith:X6SV63QHnew · submitted 2022-09-13 · 🧮 math.AG

On the finite generation of valuation semigroups on toric surfaces

Klaus Altmann , Christian Haase , Alex K\"uronya , Karin Schaller , Lena Walter This is my paper

Pith reviewed 2026-05-24 11:33 UTC · model grok-4.3

classification 🧮 math.AG
keywords valuation semigrouptoric surfacefinite generationlattice polytopenon-toric valuationample divisorcombinatorial criterionalgebraic geometry
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The pith

A combinatorial criterion decides finite generation of valuation semigroups on smooth toric surfaces for non-toric maximal-rank valuations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out a test, based only on the lattice polytope of an ample divisor and the combinatorial data of a non-toric valuation, that determines whether the associated valuation semigroup is finitely generated. A reader would care because finite generation controls whether the corresponding graded ring is Noetherian and whether certain algebraic constructions remain manageable. The authors prove the criterion works on smooth toric surfaces and then apply it to produce an explicit lattice polytope for which every such semigroup arising from one-parameter subgroups centered at a non-toric point fails to be finitely generated.

Core claim

For an ample divisor on a smooth toric surface and a non-toric valuation of maximal rank, finite generation of the valuation semigroup is equivalent to a purely combinatorial condition on the polytope and the valuation. The condition can be checked directly from the fan and the polytope without additional geometric data. As a consequence, there exists a lattice polytope such that none of the valuation semigroups coming from one-parameter subgroups centered at a non-toric point are finitely generated.

What carries the argument

The combinatorial criterion that reduces finite generation of the valuation semigroup to explicit checks on the lattice polytope and the ray of the valuation.

If this is right

  • When the criterion holds, the valuation semigroup is finitely generated and the associated graded ring is Noetherian.
  • When the criterion fails, the semigroup is not finitely generated.
  • There exist polarized toric surfaces for which every one-parameter-subgroup valuation centered at a non-toric point yields an infinitely generated semigroup.
  • The same combinatorial data decides the question uniformly for all such valuations on a given polarized toric surface.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The criterion isolates the effect of moving the valuation center off the torus, showing that non-toric points can systematically destroy finite generation.
  • One can now enumerate many polytopes and valuations to map exactly which polarized toric surfaces admit at least one finitely generated non-toric semigroup.
  • The construction supplies explicit infinite families of examples that can be used to test conjectures about when Rees algebras or associated graded rings remain finitely generated.

Load-bearing premise

The surface must be smooth and toric, the divisor ample, and the valuation non-toric of maximal rank so that the finite-generation question reduces to combinatorial data of the polytope alone.

What would settle it

Take a concrete smooth toric surface, an ample divisor, and a non-toric maximal-rank valuation; compute the semigroup generators by hand or machine and check whether they are finite precisely when the combinatorial criterion predicts they are.

Figures

Figures reproduced from arXiv: 2209.06044 by Alex K\"uronya, Christian Haase, Karin Schaller, Klaus Altmann, Lena Walter.

Figure 1
Figure 1. Figure 1: The good 7-gon ∆(D). (A) The polytope ∆(D) and the rational line segment ∆(D) vN . (B) The normal fan Σ of ∆(D) together with the two cones σ + ∋ vN and σ − ∋ −vN . As an application of the above theory, we construct in Example 6.11 a lattice polygon with a strong non-finite-generation property. To be more concrete, we look at the ample divisor D associated with the polytope ∆(D) given in Figure 11A on the… view at source ↗
Figure 2
Figure 2. Figure 2: B) ∆ nef = conv([0,0],[−7,0],[−3,−2],[0,−2]). 2.4. An alternative view on ∆ nef Beside the explicit description of Lemma 2.1, it is possible to describe the shape of ∆ nef in the following more combinatorial way. The relation vM ∈ v ⊥ N among our curve parameters means ⟨0,vN ⟩ = ⟨vM,vN ⟩ = 0; i.e., ∆ newt = conv(0,vM) is contained in the level set [vN = 0]. ρ0 ρ3 ρ2 ρ1 vN Σ (A) ∆(D) ∆newt ∆nef (B) [PITH_F… view at source ↗
Figure 3
Figure 3. Figure 3: A). Moreover, we define σmax, σmin to be the two-dimensional cones generated by the two edges of ∆(D) that contain the vertices rmax and rmin, respectively. We take the line segment ∆ newt and fit it inside the cone σmax until it hits both rays of this cone. In this way, we construct a lattice triangle ∆max with base ∆ newt and top vertex rmax. We construct ∆min (cf. Figure 3A) in a similar way. In other w… view at source ↗
Figure 4
Figure 4. Figure 4: Projection map. π: M !! M = M/Z[−3,−2], [x,y] 7! −2x + 3y [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Alternative view on Θ(ℓ,k). (A) The Minkowski sum Θ(ℓ,k)+k∆ nef and Θ(ℓ,k)+k∆ newt . (B) The cut of ∆(D) along rmax ′ +k∆ newt and rmin ′ +k∆ newt into □max, □min, and ∆(D) C with ∆(D) C = Θ(ℓ,k) + k∆ newt . Example 3.4. Continuing Example 3.2, Figure 5A shows that the inclusions Θ(ℓ,k) + k∆ nef = conv([0,0],[−8,0],[−3,−5/2],[0,−5/2]) ⊊ ∆(D) and Θ(ℓ,k) + k∆ newt = conv([0,0],[−1,0],[−4,−2],[−3,−5/2],[0,−1/… view at source ↗
Figure 6
Figure 6. Figure 6: Projected polytopes. Recall that the (torus-equivariant) global sections of L ′ (ℓ,k) are encoded by the elements of Θ(ℓ,k)∩ M. Under this identification, their pullbacks via ι ∗ are given by their images under π. Denote their number by e(ℓ,k) := #π(Θ(ℓ,k)∩ M). Summarizing what we have done so far, we obtain the following. Proposition 3.8. The pullback ι ∗L ′ (ℓ,k) = OP1 (Ξ(ℓ,k)) is a line bundle on P1 of … view at source ↗
Figure 7
Figure 7. Figure 7: Newton–Okounkov body ∆Y• (D) with flipped coordinates. This example already gives an instance of a vertex that does not lift to the semigroup (cf. Definition 5.3) when building the Newton–Okounkov body in question. Let us consider the vertex [ 2 3 , 35 3 ] and fix ℓ = 3, k = 2. The respective polyhedra 3∆(D), 2∆ newt , 2∆ nef, and Θ(ℓ,k) = (3∆(D) : 2∆ nef) are pictured in [PITH_FULL_IMAGE:figures/full_fig… view at source ↗
Figure 8
Figure 8. Figure 8: To hit the vertex [ 2 3 , 35 3 ], the value of e(3,2) would have to coincide with d(3,2) = 35. However, we only obtain e(3,2) = # π(Θ(ℓ,k)∩ M)  = 30. The red lines in [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Strongly decomposable primitive element. [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The good 7-gon ∆(D). (A) The polytope ∆(D) having seven vertices together with the rational line segment ∆(D) vN = ∆(D) ∩ [vN = 3] and its two vertices v1 = [8/3,3], v2 = [9,3]. (B) The normal fan Σ of ∆(D) having seven rays ρi (0 ≤ i ≤ 6) together with the two cones σ + = cone(ρ4, ρ6) ∋ vN and σ − = cone(ρ0, ρ4) ∋ −vN . In op. cit., the authors construct examples of projective toric surfaces whose blow-u… view at source ↗
Figure 11
Figure 11. Figure 11: SY• (D) non-finitely generated for all vN . (A) The polytope ∆(D) associated with an ample divisor D on X = TV (Σ). (B) The fan Σ of TV (Σ) with 16 rays. We will use our characterization in Theorem 6.8. As ∆(D) is centrally symmetric, the longest line segment ∆(D) vN in Definition 6.7 will pass through the origin, whatever vN . We distinguish two cases: either the endpoints of the segment ∆(D) vN are vert… view at source ↗
Figure 12
Figure 12. Figure 12: Illustration of Example 6.11. (A) The line segment ∆(D) vN containing the interior of the edges e ± 1 , e± 2 . (B) The possible region for ±vN in red together with the two cones σ ± . (C) The line segment ∆(D) vN hitting two vertices v1,v2 of ∆(D). (D) The cones σ ± containing ±vN . Now, we want to define an ample torus-invariant divisor D such that Θ(ℓ,k) = (∆(D) : ∆ nef). As an intermediate step, set D′… view at source ↗
Figure 13
Figure 13. Figure 13: Illustration of Proposition 6.12. (A) The set of rays Σ ′ (1) = Σ(1)∪ {ρ4, ρ5}, a primitive element vN = (−2,3) ∈ N, and the cone σ in which vN is strongly decomposable. (B) The Newton polytope ∆ newt given by vM = [−3,−2] and the polytope ∆ nef corresponding to C ′ in X ′ = TV (Σ ′ ). (C) The polytope Θ(ℓ,k) having a vertex rmin ′ with tangent cone σ ∨. (D) The polytope ∆(D′ ) corresponding to the non-am… view at source ↗
read the original abstract

We provide a combinatorial criterion for the finite generation of a valuation semigroup associated with an ample divisor on a smooth toric surface and a non-toric valuation of maximal rank. As an application, we construct a lattice polytope such that none of the valuation semigroups of the associated polarized toric variety coming from one-parameter subgroups and centered at a non-toric point are finitely generated.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper provides a combinatorial criterion for the finite generation of a valuation semigroup associated with an ample divisor on a smooth toric surface and a non-toric valuation of maximal rank. As an application, it constructs a lattice polytope such that none of the valuation semigroups of the associated polarized toric variety coming from one-parameter subgroups and centered at a non-toric point are finitely generated.

Significance. If the criterion is valid, the work supplies an explicit combinatorial test based on the fan, polytope, and valuation data that stays internal to toric combinatorics; this is a clear strength for the field. The explicit lattice-polytope example demonstrates concrete instances of non-finite generation and thereby supplies falsifiable predictions that can be checked directly from the combinatorial input.

minor comments (2)
  1. [Abstract] The abstract states the criterion exists but does not indicate whether it is expressed purely in terms of lattice-point counts or involves additional inequalities; a single sentence clarifying the form would help readers assess applicability.
  2. Notation for the non-toric valuation and the associated polytope data should be introduced once in a dedicated preliminary subsection and then used uniformly.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report and the recommendation to accept. The summary accurately captures the main results: the combinatorial criterion for finite generation of valuation semigroups on smooth toric surfaces under non-toric maximal-rank valuations, and the explicit lattice polytope example showing non-finite generation for one-parameter subgroups centered at a non-toric point.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained in combinatorial data

full rationale

The manuscript states a combinatorial criterion for finite generation of valuation semigroups on smooth toric surfaces, derived directly from the fan, ample divisor polytope, and maximal-rank non-toric valuation data. The criterion is presented as checkable from this input alone, and the application constructs an explicit lattice polytope for which the same criterion certifies non-generation for all one-parameter subgroup valuations at the chosen point. No equation or claim reduces a derived quantity to a fitted parameter by construction, no load-bearing step rests on a self-citation chain, and the central result does not rename or smuggle an ansatz. The argument stays internal to toric combinatorics with no reduction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated or detectable.

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Reference graph

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