Dicritical divisors and hypercurvettes

Enrique Artal Bartolo; Willem Veys

REVIEW 1 major objections 4 minor 13 references

Dicritical divisors and hypercurvettes

T0 review · 1 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that for any composition of admissible blow-ups at a smooth point of a complex variety, one can prescribe exactly which exceptional components are dicritical and with which positive integer degrees, by constructing an…

desk verdict A genuine higher-dimensional dicritical-existence theorem with a real new technique and one localized but repairable gap in the key proof. read the letter →

arxiv 2505.24648 v1 pith:DDAZITMQ submitted 2025-05-30 math.AG

Enrique Artal Bartolo , Willem Veys This is my paper · ORCID

classification math.AG MSC 14E0514E1514B05

keywords dicriticaldivisorshypercurvettesrationalfunctionsblow-upsexceptionalvaluationsalgebraicsingularitiesbirationalmaps

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves a full realization theorem in arbitrary dimension. Fix a smooth point p of a complex variety and any composition of admissible blow-ups that is an isomorphism away from p. For any nonempty set J of exceptional components and any positive integers d_j, there is a rational function h whose lift restricts to a dominant map of degree d_j on E_j exactly for j in J, and to a constant map on all other components. This is the higher-dimensional analogue of the two-dimensional curvette theorem, and it solves the paper's Goal 2.5 completely.

What carries the argument

The central object is the hypercurvette, a higher-dimensional curvette: for each exceptional divisor E_i one chooses a hypersurface germ C_i through p whose strict transform meets E_i in a general hyperplane of the projective-space fiber and has normal crossings with the exceptional locus. The associated valuation matrix A=(a_{ji}) with a_{ji}=\nu_i(C_j) describes how pullbacks of the C_i decompose into exceptional divisors; its principal submatrices have determinant 1, making the relevant linear systems solvable in integers. To control the degree of the final dicritical component, the proof introduces special hypersurfaces H_j whose strict transforms intersect E_s along E_s\cap E_j plus fibers, compensating the contributions of earlier hypercurvettes. The final step adds high powers of hypercurvettes of later components and uses the inequalities of Steps 5-7 to keep all non-dicritical components constant while preserving the prescribed degree.

What would settle it

For a concrete admissible sequence in dimension 3, choose a family of hypercurvettes and compute the valuation matrix; if some principal submatrix has determinant different from 1, the starting point of the proof collapses. Alternatively, in the explicit setting of Example 4.6, take k=5 and ell=13 and verify that the restriction of the constructed function to E_4 is constant; if it is nonconstant, the interval inequalities in the final step of the proof of Theorem 4.4 do not hold.

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Extended reading notes

Core claim

Theorem 1: for any nonempty J in {1,...,m} and any positive integers d_j, there exists a rational function h for which E_j is dicritical of degree d_j if j is in J, and E_j is non-dicritical otherwise. The proof first obtains, for each single index s, a rational function whose only dicritical component is E_s with prescribed degree (Theorem 4.4); then generic Mobius transformations of such functions are multiplied to impose simultaneously the prescribed behavior on every component (Proposition 4.8). The construction is explicit, built from products of equations of hypercurvettes and of auxiliary special hypersurfaces.

Load-bearing premise

The load-bearing premise is that every ordered sequence of admissible blow-ups admits hypercurvettes with normal-crossing behavior and a valuation matrix whose principal submatrices all have determinant one; if this structural input ever fails, the linear-algebra construction of the rational function cannot get off the ground.

Editorial extensions

If this is right

For any admissible blow-up sequence at a smooth point, the exceptional components can be assigned any prescribed pattern of dicritical and non-dicritical behavior with any prescribed positive degrees.
The two-dimensional curvette theorem becomes the n=2 case of Theorem 1, where hypercurvettes reduce to ordinary curvettes and the valuation matrix is canonical and symmetric.
The proof is constructive: the desired rational function is produced as a quotient of products of hypercurvette equations and special-hypersurface equations, with generic coefficients chosen at each step.
Combining single-dicritical functions through Mobius transformations preserves dicritical behavior and degree, so arbitrary subsets of simultaneous dicritical components are realized from the one-dicritical case.
The theorem resolves the stated higher-dimensional realization problem without imposing restrictions on the dimension, the length of the blow-up sequence, or the positions of the centers.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The proof really needs only the unimodularity of all principal submatrices of the valuation matrix together with hypercurvette normal-crossing behavior; any modification supplying such a matrix system would satisfy the same theorem, so the phenomenon seems tied to a matrix property rather than specifically to smoothness of the ambient variety.
Because Example 1.2 shows that the degree of a dicritical component can depend on the order of blow-ups even when the final modification is the same, one could ask for a classification of achievable degree-vectors over all admissible orders for a fixed final modification; the paper does not address that classification.
A natural stress test is to run the same construction with a singular base point or non-smooth admissible centers; the argument would survive wherever hypercurvette existence and the unimodularity condition can still be established, and fail exactly where they cannot.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper proves the expected higher-dimensional version of the Abhyankar–Artal theorem: for any modification of a smooth point by admissible blow-ups, you can prescribe exactly which exceptional divisors are dicritical and with what degree. That is a real, nontrivial generalization, and the proof is mostly solid. The new tool is the special hypersurfaces of Section 3, which compensate for the fact that hypercurvettes in dimension ≥3 meet many exceptional components. That was the main obstruction, and the authors handle it honestly.

The linear-algebra structure is clean: the valuation matrix of hypercurvettes is unimodular in its principal submatrices (from [13]), and that is what makes the systems solvable. The examples in Sections 1 and 4 are genuinely useful; Example 1.2 showing order-dependence of degree is nice. The paper is well-written.

The soft spot is Step 7 in the proof of Theorem 4.4. The text says 'recall that a_{si}=a_ss'. That is false when a later center sits in several exceptional divisors; the correct identity is a_{si}=p_i a_ss, with p_i as defined in the same step. The stress-test example is right: in C^3 with four blow-ups, a_{1,4}=2 while a_{11}=1 and p_4=2. The good news is that the intended conclusion still follows if you replace the false identity with the correct one: inequality (5.8) gives ℓ a_{si} < α_i just as well. So the gap is localized and repairable, but as printed the proof of Theorem 4.4 is not complete. The authors should fix it.

The other soft spot is minor: Step 5's 'big enough' choices are existence claims without explicit bounds. That is acceptable for this type of proof, and they do give a rough bound.

The paper relies on the earlier hypercurvette machinery from [13]; that is independent published work, not circular. I do not see any fitting or self-referential shortcuts. All told, this is a serious contribution in a narrow but honest subfield.

I would send it to a competent referee. It deserves revision, not rejection. If the authors fix the a_{si} point, I would accept it.

Referee Report

1 major / 4 minor

Summary. The paper proves Theorem 1: given a modification of a smooth variety obtained by a sequence of admissible blow-ups at a smooth point p, and any prescribed subset J of exceptional components with assigned positive degrees d_j, there exists a rational function h for which precisely the components in J are dicritical, with the prescribed degrees. The proof builds on the second author's prior notion of higher-dimensional curvettes (hypercurvettes), whose associated valuation matrix has unimodular principal submatrices, and introduces new 'special hypersurfaces' to compensate for uncontrolled intersections. The main technical result is Theorem 4.4, whose proof occupies Section 5. Proposition 4.8 then combines the one-dicritical-component case to obtain the general statement.

Significance. If correct, this settles the natural higher-dimensional analogue of the Abhyankar--Artal classification of dicritical components, with full control of prescribed degrees for arbitrary sequences of admissible blow-ups. The paper is well structured and readable, with worked examples illustrating the new phenomena (Examples 1.2, 3.4, 4.2, 4.5, 4.6). It is also transparent about its dependency on the published hypercurvette construction and unimodularity result in [13]. The reduction of the problem to a unimodular linear system in Proposition 2.8 is elegant, and the special hypersurface technique in Lemma 3.1 is a useful new tool. The main theorem is significant and the proof strategy is sound in its overall architecture, but one step in the central proof contains a false identity that must be repaired.

major comments (1)

[Section 5, Step 7 (proof of Theorem 4.4)] The sentence 'recall that a_{si}=a_ss' is false in general. The recursion in Example 2.3, Eq. (2.1), gives a_{si} = sum_{a in A_i} a_{sa} for s < i, and since p_i is defined in Step 7 by the same recursion (with implicit base p_{s+1}=1), one obtains a_{si} = p_i a_ss, not a_{si}=a_ss. For instance, in C^3, after blowing up the origin, then a point in E_1, then a conic in E_2 avoiding E_1 cap E_2, then a point in E_2 cap E_3, one has s=1, i=4, a_{1,4}=2, a_{1,1}=1, and p_4=2. Because the false identity is used to conclude N_i>0 from condition (5.8), the written proof of Theorem 4.4 is incomplete. The gap is local and repairable: replacing the equality by a_{si}=p_i a_ss, the inequality in (5.8) gives l a_{si} = l p_i a_ss < p_i(alpha_s + a_ss W_i) = alpha_i, so N_i>0 follows; the repair requires inserting the short induction proving a_{si}=p_i a_ss. Since Theorem 4.4 is the load-bearing step for Theorem 1, this is a genuine gap in the printed proof rather than a typographical slip.

minor comments (4)

[Section 5, after Eq. (5.3)] The statement 'Es is dicritical of degree 1 for h' should read 'degree d', since the construction in (4.1) uses (C'_s)^d/(C''_s)^d and Theorem 4.4 is stated for an arbitrary positive degree d.
[Section 5, Steps 6 and 7] The integer p_{s+1} is used implicitly in Step 7 (the induction base) but never defined; the proof should explicitly set p_{s+1}=1, for example in Step 6.
[Section 5, Step 5] The symbol k is used both for the number of special hypersurfaces from Proposition 4.1 and later for the vector (k_{s+1}, ..., k_m) of exponents in (5.1). This overloaded notation is confusing and should be resolved, e.g., by renaming the integer k in Proposition 4.1 as r or K.
[Section 4, proof of Proposition 4.1] The choice of the special hypersurfaces H_j 'still with the normal crossing restriction' is asserted but not justified; the manuscript would benefit from a short genericity argument showing that the H_j from Lemma 3.1 and Lemma 3.6 can be chosen so that their strict transforms together with all C_i and E_i form a normal crossing divisor in X_s.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the construction is self-contained given independent prior hypercurvette results; the target theorem is not assumed.

full rationale

The central claim is an existence theorem for rational functions with prescribed dicritical exceptional divisors and degrees, and the proof chain is constructive. Proposition 1.3 supplies hypercurvettes from the second author's earlier paper [13], and Proposition 1.7 supplies the unimodularity of the valuation matrix A; these are external published structural results whose assumptions (an ordered modification by admissible blow-ups) do not include the target statement, so they count as independent support rather than circularity. Proposition 2.8 solves a linear system rA=N using unimodularity, with no quantity fitted to data and later renamed a prediction. Proposition 4.1 builds h with the factor (C'_s)^d/(C''_s)^d, so the prescribed degree is deliberately engineered into the construction; this is a legitimate existence proof, not a self-definitional or fitted-input step. Theorem 4.4 and Section 5 then adjust parameters k and ell by explicit inequalities, derived from the valuation recursion in Lemma 2.2, to keep earlier components non-dicritical and make later components non-dicritical; the inequalities are not obtained from the desired conclusion. I found no equation that is equivalent to its input by construction, no load-bearing self-citation chain that forces the result, and no renaming of a known empirical pattern as an organizational principle. The possible technical issue in Step 7 of Section 5, where the identity a_{si}=a_ss is invoked, concerns correctness of the printed proof rather than circularity, so it does not affect this score.

Assumptions & free parameters 0 free parameters · 3 assumptions · 2 invented entities

The central construction rests on the second author's hypercurvette theory: the existence of hypercurvettes for any admissible sequence of blow-ups and the unimodularity of their valuation matrix. These are imported from [13], and the rest of the proof is a combination of standard algebraic geometry and the new special-hypersurface technique developed in Section 3.

assumptions (3)

domain assumption Existence of hypercurvettes C_j with normal crossing properties for any ordered modification (Prop 1.3).
Imported from [13, Prop 3.2]; the paper does not reprove it. It is the starting point for the valuation matrix and all constructions.
domain assumption The principal submatrices A_t of the valuation matrix have determinant 1 (Prop 1.7).
Imported from [13, Prop 3.3]. Unimodularity is what makes the linear systems in Propositions 2.8 and 4.1 solvable over the integers.
standard math Standard algebraic geometry facts: blow-ups, strict transforms, Bertini genericity, normal crossing divisors.
Used throughout, for example in Lemma 3.1 and Section 5; these are standard and accepted.

invented entities (2)

Special hypersurfaces H_j independent evidence
purpose: To compensate the unwanted intersections of hypercurvettes of earlier exceptional components with the dicritical divisor E_s, enabling the degree prescription.
They are constructed in Lemma 3.1 and refined in Lemma 3.6; their existence and intersection properties are proved within the paper, and they are explicitly used in Proposition 4.1 and Theorem 4.4.
Hypercurvettes (higher-dimensional curvettes) C_j independent evidence
purpose: Generalizations of 2D curvettes to higher dimensions; they provide hypersurfaces whose valuation matrix is unimodular.
Introduced in [13] with properties recalled in Proposition 1.3; the present paper relies on their existence and normal crossing behavior.

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Pith. "Pith review of Dicritical divisors and hypercurvettes." pith.science (2026). https://pith.science/paper/DDAZITMQ

@misc{pith2026250524648,
  author       = {Pith},
  title        = {Pith review of: Dicritical divisors and hypercurvettes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DDAZITMQ}},
  note         = {Machine review of arXiv:2505.24648}
}

abstract

Germs of rational functions~$h$ on points $p$ of smooth varieties~$S$ define germs of rational maps to the projective line. Assume that $p$ is in the indeterminacy locus of $h$. If $\pi:\hat{S}\to S$ is a birational map which is an isomorphism outside $p$, then $h$ lifts to a germ of a rational map on $(\hat{S}, \pi^{-1}(p))$. The exceptional components $E_i$ of $\pi^{-1}(p)$ are classified according to the restriction of (the lift of) $h$ to $E_i$; the dicritical components are those where this restriction induces a dominant map. In a series of papers, Abhyankar and the first named author studied this setting in dimension $2$, where the main result is that, for any given $\pi$, there is a rational function $h$ with a prescribed subset of exceptional components that are dicritical of some given degree. The concept of curvette of an exceptional component played a key role in the proof. The second named author extended previously the concept of curvette to the higher dimensional case. Here we use this concept to generalize the above result to arbitrary dimension.

Figures

Figures reproduced from arXiv: 2505.24648 by the authors.

**Figure 1.** π2 ◦ π1 We take as X3 π3 −→ X2 the blow-up along L. We summarize the result: • E3 ⊂ X3 is isomorphic to Σ2, and ℓ3 is a general fiber; • E2 ⊂ X3 is isomorphic to the blow-up of a point in E2 ⊂ X2, i.e., it is isomorphic to Σ1. The class ℓ2 is a general (+1)-section in Σ1. Note that E23 := E2 ∩ E3 is the (−1)-section of E2 and a fiber in E3; • E1 ⊂ X3 is isomorphic to Σ1 and ℓ1 is a general (+1)-section. Note that E1… view at source ↗

**Figure 2.** π¯2 ◦ π¯1 Let π¯3 : X¯ 3 → X¯ 2 be the blow-up along F. It is not hard to check that X3 = X¯ 3, and that in X3 = X¯ 3 we have E1 = E¯ 1 and Ej = E¯ k, {j, k} = {2, 3}. Moreover, we have ¯ℓ1 = ℓ1 and ¯ℓ2 = ℓ3 (as classes) but ℓ2 is a general (+1)-section while ¯ℓ3 is a fiber of E¯ 3 = E2 ∼= Σ1, see [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** X3 = X¯ 3 Proposition 1.3 ([13, Proposition 3.2]). Consider the ordered modification π as in (1.1). One can construct consecutively for j = 1, . . . , m a hypersurface Cj on X0 with the following properties. (1) The strict transform of Cj in Xj−1 contains Zj and is smooth along Zj . (2) Denoting by C˜ i , i ≤ j, the strict transform in Xj of Ci , we have that E1∪E2∪· · ·∪Ej∪C˜ 1∪C˜ 2∪· · ·∪C˜ j is a normal crossing … view at source ↗

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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