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arxiv: 2312.01722 · v2 · pith:SUWJGF67new · submitted 2023-12-04 · 🧮 math.AG · math.NT

Local Euler characteristics of A_n-singularities and their application to hyperbolicity

Nils Bruin , Nathan Ilten , Zhe Xu This is my paper

Pith reviewed 2026-05-24 05:01 UTC · model grok-4.3

classification 🧮 math.AG math.NT
keywords local Euler characteristicA_n singularitiessymmetric differentialsquasi-hyperbolic surfacestoric geometrylattice point enumerationalgebraic surfaces
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The pith

The local Euler characteristic of the mth symmetric power of the cotangent bundle on an A_n singularity is a quasi-polynomial in m of period n+1.

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

The paper computes the local Euler characteristic for sheaves of symmetric differentials on isolated A_n surface singularities using toric geometry. It derives an explicit quasi-polynomial formula in the power m with period n+1, also expressible as lattice point counts in a polyhedron. Applying this to a family of surfaces with many singularities shows that members of degree 8 or higher have no rational curves, while those of degree 10 or higher also lack elliptic curves. This produces new examples of low-degree quasi-hyperbolic surfaces in projective three-space.

Core claim

For an isolated surface singularity of type A_n, the local Euler characteristic of the mth symmetric power of the cotangent bundle is a quasi-polynomial in m of period n+1. This quantity can also be expressed as the number of lattice points in a certain non-convex polyhedron. When summed over the singularities on explicit surfaces constructed by Labs, the resulting bounds imply the absence of genus-zero curves on surfaces of degree at least eight and the absence of both genus-zero and genus-one curves on those of degree at least ten.

What carries the argument

The local Euler characteristic of the mth symmetric power of the cotangent bundle on an A_n singularity, computed via toric geometry as a quasi-polynomial or lattice-point count.

If this is right

  • Summing local terms controls the existence of low-genus curves on the global surface.
  • The Labs family yields algebraic quasi-hyperbolic surfaces in P^3 starting at degree eight.
  • The periodicity allows efficient computation for large m.

Where Pith is reading between the lines

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

  • Similar computations could apply to other classes of singularities to find more hyperbolic examples.
  • The lattice-point interpretation might connect to other enumerative problems in toric varieties.
  • Verifying the formula for small values of n and m provides a check on the toric method.

Load-bearing premise

The local Euler characteristic computations via toric geometry and lattice-point enumeration apply directly to the isolated A_n singularities on the specific surfaces constructed by Labs, allowing the local terms to be summed to control global curve existence.

What would settle it

The discovery of a rational curve on a degree-eight member of the Labs family of surfaces would falsify the hyperbolicity claim.

Figures

Figures reproduced from arXiv: 2312.01722 by Nathan Ilten, Nils Bruin, Zhe Xu.

Figure 2.1
Figure 2.1. Figure 2.1: The cone and dual cone for an An-singularity The above construction globalizes. Let Σ be a fan in N ⊗ R, that is, a collection of pointed polyhedral cones that is closed under taking faces, and such that any two elements intersect in a common face. Any face relation τ ≺ σ for σ ∈ Σ induces an open inclusion Xτ ,→ Xσ . The toric variety XΣ is constructed by gluing together the affine toric varieties {Xσ }… view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: The minimal resolution of an An-singularity that is, VF is the k-vector space obtained as the T -invariant sections of the restriction of F to the torus T . The restriction of F to T is a vector bundle, and VF may be identified with the fibre of this bundle over the identity element of T . In particular, it is a vector space of dimension equal to the rank of F . For each ray ρ ∈ Σ (1), we may consider th… view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: The fans eΣ, Σ, and Σ The surface XΣ has a single An-singularity (see Example 2.1). The surface XΣ has a single An−1-singularity: this may be seen by applying the lattice isomorphism [PITH_FULL_IMAGE:figures/full_fig_p011_3_1.png] view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Regions of linearity of δn(m,u) χu(S mΩˆ XΣ ) and cancelling terms, we obtain δn(m,u) = dimW ρ1 (u) − dimW σ01 (u) − dimW σ1(n+1)(u) + dimW σ0(n+1)(u) = −dimV ρ1 (ρ1(u)) + dim(V ρ0 (ρ0(u)) + V ρ1 (ρ1(u))) + dim(V ρ1 (ρ1(u)) + V ρn+1 (ρn+1(u))) − dim(V ρ0 (ρ0(u)) + V ρn+1 (ρn+1(u))) = (m + 1) − λm(ρ1(u)) − max{m + 1 − λm(ρ0(u)) − λm(ρ1(u)),0} − max{m + 1 − λm(ρ1(u)) − λm(ρn+1(u)),0} + max{m + 1 − λm(ρ0(u)… view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: Lattice points of the region Π(C1) We are now able to use induction to obtain a formula for χ(n,m) as a weighted lattice point count. Using notation introduced at the start of this subsection, define ∆n = 2 ∗  1 n + 1 : 2(n + 1) − 1 n + 1 + n ∗ γ. Theorem 3.5. For n,m ≥ 1 we have χ(n,m) = X (x,y)∈((m+1)·∆n)∩Z2 y. Proof. Up to integral translation in the x-direction and reflection around the line x = 0,… view at source ↗
Figure 3.4
Figure 3.4. Figure 3.4: Example arrays from the proof of Lemma 3.9 For [z m]g(z) we use the form of g(z) from Lemma 3.9 and obtain that g(z) is equal to X k≥0 q=0,...,n [PITH_FULL_IMAGE:figures/full_fig_p018_3_4.png] view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Top view of G0,3 along with the half-open convex polytopes Pi = Conv{Pi−1,Qi−1,Pi ,Qi ,Z} \ Conv{Pi ,Qi ,Z} \ Conv{Pi−1,Pi ,Z}, Cn = Conv{Pn,τn(Pn),Qn,τn(Qn),Z} \ Conv{Pn,τn(Pn),Z} from (1.4). For reference we record P ′ n = τn(Pn) =  1 n + 1 ,−1,0  , Q ′ n = τn(Qn) = 2 (n + 1)(n + 2),−1, n n + 2! . Lemma 4.3. The non-convex polytope Gm,n is the dilation by m + 1 of G0,n. Furthermore, we have G0,n = Cn… view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Intersection of (m + 1)Cn with a = 0 (1) First we note that lattice point counts are non-decreasing with increasing dilation, so χ 0 (sn, SmΩ1 Y ) is non-decreasing in m. Since Gn,m = (Gn,m ∩ H)∪τn(Gn,m ∩ H) and τn(Z3 ) = Z3 , we see from the observation above that the lattice point count is also non-decreasing in n. (2) To establish that χ 0 (sn, SmΩ1 Y ) is constant in n for n > m, we observe that (m+1… view at source ↗
read the original abstract

Wahl's local Euler characteristic measures the local contributions of a singularity to the usual Euler characteristic of a sheaf. Using tools from toric geometry, we study the local Euler characteristic of sheaves of symmetric differentials for isolated surface singularities of type $A_n$. We prove an explicit formula for the local Euler characteristic of the $m$th symmetric power of the cotangent bundle; this is a quasi-polynomial in $m$ of period $n+1$. We also express the components of the local Euler characteristic as a count of lattice points in a non-convex polyhedron, again showing it is a quasi-polynomial. We apply our computations to obtain new examples of algebraic quasi-hyperbolic surfaces in $\mathbb{P}^3$ of low degree. We show that an explicit family of surfaces with many singularities constructed by Labs has no genus $0$ curves for the members of degree at least $8$ and no curves of genus $0$ or $1$ for degree at least $10$.

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

1 major / 2 minor

Summary. The paper computes Wahl's local Euler characteristic for the m-th symmetric power of the cotangent bundle on isolated A_n surface singularities via toric geometry. It proves an explicit quasi-polynomial formula in m of period n+1 and an equivalent expression as the number of lattice points in a non-convex polyhedron. These local computations are applied to an explicit family of singular surfaces in P^3 constructed by Labs, yielding that the surfaces have no genus-0 curves for degree at least 8 and no curves of genus 0 or 1 for degree at least 10.

Significance. If the derivations hold and the local-to-global summation applies without additional terms, the explicit quasi-polynomial and lattice-point formulas provide concrete tools for controlling symmetric differentials near A_n singularities. The application produces new explicit examples of algebraic quasi-hyperbolic surfaces in P^3 of low degree, which is of interest for questions on hyperbolicity and curve existence on surfaces.

major comments (1)
  1. [Application section] Application section: the claim that the summed local Euler characteristics control the existence of low-genus curves on the Labs surfaces requires that every singularity is an isolated A_n singularity of the exact local analytic type for which the toric resolution and non-convex polyhedron are derived, with no extra contributions from the global embedding in P^3. The manuscript should include an explicit verification or reference confirming the singularity types and isolation for each member of the family (at least for degrees 8 and 10).
minor comments (2)
  1. [Toric geometry setup] The notation for the polyhedron whose lattice points count the components of the local Euler characteristic could be introduced with a diagram or explicit coordinate description earlier in the toric-geometry setup to improve readability.
  2. [Main formula] A brief comparison table of the new quasi-polynomial formula against previously known cases (e.g., for small n or m) would help readers verify the period n+1 claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for the constructive comment on the application section. We address it point by point below.

read point-by-point responses

  1. Referee: [Application section] Application section: the claim that the summed local Euler characteristics control the existence of low-genus curves on the Labs surfaces requires that every singularity is an isolated A_n singularity of the exact local analytic type for which the toric resolution and non-convex polyhedron are derived, with no extra contributions from the global embedding in P^3. The manuscript should include an explicit verification or reference confirming the singularity types and isolation for each member of the family (at least for degrees 8 and 10).

    Authors: We agree that an explicit confirmation strengthens the application. Labs' original construction (referenced in the manuscript) explicitly produces surfaces whose singularities are isolated A_n singularities of the precise local analytic type used in our toric resolution and non-convex polyhedron count. In the revised manuscript we will add a short paragraph in the application section that (i) cites the relevant statements from Labs' work confirming the singularity types and isolation for the family members of degree 8 and 10, and (ii) notes that, because the singularities remain isolated, the global embedding in P^3 contributes no additional terms to the local Euler characteristic beyond those already accounted for by our local formulas. This verification is straightforward from the cited reference and does not require new computations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation from toric geometry is self-contained

full rationale

The paper computes the local Euler characteristic of symmetric powers of the cotangent bundle on A_n singularities via toric geometry and lattice-point counts in a non-convex polyhedron, yielding an explicit quasi-polynomial formula of period n+1. This is a direct mathematical derivation from standard methods applied to the local analytic type, not a fit to data or a renaming of known results. The subsequent application sums these local terms over singularities on Labs' surfaces to bound global sections and conclude absence of low-genus curves. No steps reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the central claims rest on independent toric computations that do not presuppose the hyperbolicity conclusion. This is the normal case of a self-contained algebraic geometry argument.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no free parameters, axioms, or invented entities are introduced; the work relies on standard tools from toric geometry and Wahl's definition of local Euler characteristic.

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Reference graph

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