The ordinary double point as the largest singular volume

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Family 037 · Algebraic and complex geometry
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-12

1. The question

A smooth point and a singular point can belong to the same algebraic variety, yet their small neighborhoods have very different geometry. Can one number distinguish the smooth case and then identify the least severe singularity?

Normalized volume is designed to measure a singularity through all possible ways of assigning orders of vanishing to local functions. In dimension $n$, its largest value is $n^n$, attained exactly at smooth points. The ordinary-double-point gap conjecture predicts the next maximum among singular points.

Family 037 contains two manuscripts. M2 claims the sharp four-dimensional case, and M1 uses it as a base case for the theorem in every dimension at least two. The proposed maximizer is an ordinary double point: a nondegenerate quadratic cone. The statement ranges over all boundary-free complex algebraic klt germs, not just singularities already known to be hypersurfaces.

2. Background and essential definitions

Local geometry and the simplest cone

A germ $(X,x)$ means the geometry of $X$ near a chosen point $x$, ignoring what happens far away. Smoothness means the germ has local coordinates like an open subset of $\mathbb C^n$. A singular point fails this condition.

An ordinary double point in complex dimension $n$ has analytic local equation

$$z_0^2+z_1^2+\cdots+z_n^2=0 \quad\text{inside }\mathbb C^{n+1}.$$

The quadratic form is nondegenerate: its matrix has full rank. There are $n+1$ ambient coordinates and one equation, so the hypersurface has dimension $n$. Analytic equivalence allows a local holomorphic change of coordinates, which is stronger information than merely sharing a numerical invariant.

The embedding dimension is the number of independent first-order local coordinates required. Algebraically, if $\mathfrak m_x$ is the ideal of functions vanishing at $x$, it is

$$\operatorname{edim}(X,x)=\dim_{\mathbb C}\mathfrak m_x/\mathfrak m_x^2.$$

At a smooth $n$-dimensional point it is $n$; at an ordinary double point it is $n+1$.

Valuations and volumes

A real valuation $v$ assigns an order of vanishing to local functions, with

$$v(fg)=v(f)+v(g),\qquad v(f+g)\geq\min\{v(f),v(g)\}.$$

It is centered at $x$ when it is nonnegative on the local ring and positive on its maximal ideal. Ordinary vanishing order at a smooth point is a basic example; other valuations inspect weighted directions or exceptional divisors on birational models.

For $t>0$, the valuation ideal

$$\mathfrak a_t(v)=\{f\in\mathcal O_{X,x}:v(f)\geq t\}$$

contains functions vanishing to at least that order. The quotient by this ideal records the finite amount of local function data below the threshold. Its length is its dimension as a complex vector space. The asymptotic growth defines

$$\operatorname{vol}(v)=\lim_{t\to\infty} \frac{n!\,\operatorname{length}(\mathcal O_{X,x}/\mathfrak a_t(v))}{t^n}.$$

This is not the Euclidean volume of a ball. It is a normalized count of local algebraic data.

Discrepancy supplies the normalization

The log discrepancy $A_X(v)$ measures how a canonical form changes under the birational viewpoint encoded by $v$. For a prime divisor $E$ on a resolution $\pi:Y\to X$,

$$A_X(\operatorname{ord}_E)=1+\operatorname{ord}_E(K_Y-\pi^*K_X).$$

The canonical divisor must be $\mathbb Q$-Cartier so this pullback makes sense. The klt condition means these discrepancies are positive for the relevant divisors. It restricts the severity of the singularity. The papers use zero boundary, so no extra weighted hypersurfaces are included in this calculation.

Normalized volume is

$$\widehat{\operatorname{vol}}(x,X) =\inf_{v\text{ centered at }x}A_X(v)^n\operatorname{vol}(v).$$

Multiplying $v$ by a positive constant multiplies discrepancy by that constant and divides volume by its $n$-th power. Their product is unchanged. Taking the infimum therefore compares genuinely different ways of viewing the germ, rather than arbitrary choices of units for an order of vanishing.

3. Development history and previous partial results

Chi Li introduced normalized volume to connect singularity theory with K-stability and canonical metrics. The motivating metric picture concerns limits of Kähler–Einstein spaces, where tangent cones encode small-scale geometry. Spotti–Sun formulated the ordinary-double-point gap in that setting and its algebraic version for arbitrary klt germs.

Low dimensions supplied evidence and complete cases. Surface quotient calculations give normalized volume $4/|G|$ for the relevant quotient by a finite group $G$. Liu–Xu’s threefold work established the unrestricted three-dimensional gap and equality characterization. Liu later treated local complete intersections in all dimensions. These results already settle the conjecture when a sufficiently restrictive local equation is known.

Toric singularities, built from combinatorial cone data, admit a different sharp bound involving $(16/27)n^n$. Smooth polarized Fano cones also have earlier results. Neither setting covers every isolated klt singularity, so one cannot begin the general argument by assuming a convenient cone presentation or smooth quotient base.

A second line of work made the infimum itself accessible. Blum proved existence of minimizing valuations; Xu proved their quasi-monomial structure; Xu–Zhuang established uniqueness up to scaling and finite generation. The resulting stable-degeneration theory replaces a germ by an algebraic cone while retaining its normalized volume. M1 and M2 cite these primary results in their introductions and explain which versions they use.

The remaining issue is structural: a volume-preserving degeneration is not automatically a hypersurface. M2 aims to force small embedding dimension from a sufficiently large normalized volume. M1 then develops the higher-dimensional estimate and returns the equality information to the original germ.

4. What the new manuscripts claim

The sharp local gap

M1’s Theorem 1.1 states that every singular closed point of a normal complex algebraic $n$-fold, for $n\geq2$, with $\mathbb Q$-Cartier canonical divisor and klt singularities near the point, satisfies

$$\widehat{\operatorname{vol}}(x,X)\leq2(n-1)^n.$$

Equality holds precisely when the analytic germ is the nondegenerate quadratic hypersurface displayed above. The point need not be isolated in the original singular locus, and no complete-intersection, quotient, or smoothability assumption is imposed.

The comparison with smooth points is easy to display:

Complex dimension	Smooth maximum	Claimed singular maximum
2	4	2
3	27	16
4	256	162

These are normalized-volume values under the log-discrepancy convention used in the manuscripts.

M2’s four-dimensional structural statement

M2 proves the stated fourfold bound $162=2\cdot3^4$, with the same equality case and unrestricted boundary-free scope. Its central reduction claims

$$\widehat{\operatorname{vol}}(x,X)\geq162 \quad\Longrightarrow\quad\operatorname{edim}(X,x)\leq5.$$

For a singular normal fourfold, that forces a hypersurface presentation. The established complete-intersection theorem then supplies the bound and rigidity. Thus the new structural claim is what permits an earlier theorem to finish the argument.

M1 uses M2 as its sole companion input, alongside the cited established results. Its higher-dimensional proof is not independent of the fourfold paper.

A global consequence

A $\mathbb Q$-Fano variety is normal, projective, and klt, with ample anticanonical divisor $-K_V$. K-semistability is a condition controlling how its geometry behaves under algebraic degenerations; it is central to the existence of Kähler–Einstein metrics. The relevant global size is the top intersection number $(-K_V)^n$.

Combining the local gap with the established local-to-global inequality, M1 obtains, for singular K-semistable $\mathbb Q$-Fano varieties,

$$(-K_V)^n\leq2\left(\frac{n^2-1}{n}\right)^n<2n^n.$$

The paper does not claim that this smaller singular bound is sharp. Together with Li–Miao’s smooth theorem, it states that a K-semistable $\mathbb Q$-Fano variety other than projective space has anticanonical volume at most $2n^n$, with equality exactly for a smooth quadric or $\mathbb P^1\times\mathbb P^{n-1}$. The smooth equality classification is credited to that prior theorem.

5. Why the advance matters

Importance: 4/5, provisional. A sharp local invariant would single out the ordinary double point among every singularity in a broad class, without assuming a local equation. This connects asymptotic algebra, birational geometry, and metric degeneration theory.

The equality statement is as important as the inequality. A bound says how large the invariant can be; rigidity identifies the entire analytic germ when that bound is reached. The global consequence shows that a sufficiently mild-looking singularity still imposes a strict cost on the size of a K-semistable Fano variety.

The score reflects a major structural advance with several connections, while recognizing that the theorem concerns one singularity invariant under specific klt and boundary assumptions.

6. The main idea

Preserve the minimum while simplifying the germ

Both manuscripts use stable degeneration to pass to a cone with a positive grading. A grading assigns weights to functions, analogous to assigning different degrees to variables. Its minimizing real direction is a Reeb vector. The cone reduction preserves the infimum over all centered valuations, not merely the best value among a restricted family of gradings.

A large-volume hypothesis and earlier bounds constrain the cone: the difficult high-volume case becomes isolated and Gorenstein, meaning its canonical sheaf is locally a line bundle. These are conclusions of the reduction, not assumptions added to the original theorem.

Four dimensions: turn volume into few coordinates

M2 examines finite stabilizers of the grading action and counts transverse jets, finite Taylor data on slices. A stabilizer that is too large would furnish a valuation with smaller normalized volume than the minimizing one. Excluding it gives both discrepancy bounds and restrictions on the Hilbert series, the generating function that counts homogeneous functions by weight.

A Fourier-based estimate then supplies low-weight functions. Applying related estimates to suitable lower-dimensional centers controls a singularity threshold. A general hyperplane section becomes a terminal Gorenstein threefold, whose known hypersurface structure bounds the embedding dimension of the original fourfold.

This explains the logical route: high normalized volume constrains weights and thresholds; those constraints force a short local equation; the established hypersurface gap finishes the argument.

Higher dimensions: close an estimate involving a supremum

M1 separates gradings with small stabilizers from those with large stabilizers. In the first case, rational curves and tensor-weight bounds force a projective-space or quadric quotient. In the second, leading Taylor exponents form a semigroup, a set closed under addition. Positivity and adjoint lifting give enough exact control of this set to construct a comparison singularity.

The comparison has the same dimension, so ordinary induction alone would not suffice. Instead, put

$$p_n=2\left(\frac{n-1}{n}\right)^n, \qquad M_n=\sup\frac{\widehat{\operatorname{vol}}(x,X)}{n^n},$$

where the supremum ranges over singular boundary-free klt $n$-fold germs. The manuscript derives a comparison bounded by $(p_n+M_n)/2$. Applying it to values approaching the supremum forces $M_n\leq p_n$; attainment of the supremum is not assumed.

For equality, stricter counting returns the cone to the small-stabilizer case. Further local information forces a nondegenerate quadratic part in the original germ, completing the claimed analytic characterization.

7. Scope, limitations, and open questions

The theorem is for complex algebraic klt germs with zero boundary. It does not give the same formula for arbitrary singularities, positive characteristic, or pairs with nonzero boundary. Dimension one is excluded because a normal complex algebraic curve is smooth.

Normalized volume is an infimum over all centered valuations. Computing one convenient valuation only gives an upper bound and cannot by itself establish equality. The manuscripts explicitly use the minimizing and degeneration theory to address this distinction.

The singular global bound is not presented with a sharpness claim or an equality classification. More refined gaps below the ordinary-double-point value, boundary versions, and effective ways to compute the invariant remain further directions.

The conclusions above report the two manuscripts and their stated dependencies. No proof or novelty verification is part of this digest.

8. Sources and further reading

M1 §1 states the all-dimensional local and global claims and maps the prior cases; §7 explains the supremum argument and §8 treats equality. M2 §§1–2 state the fourfold theorem and structural reduction; §§3–6 develop the stabilizer, Hilbert-series, and hyperplane-section steps.

For the global connection, see Liu’s local-to-global volume work. The linked Spotti–Sun and Liu–Xu papers give primary context for the conjecture and its three-dimensional predecessor. An optional introduction to valuations supplies further examples of orders of vanishing.

Source manuscripts

• M1. OpenAI, The ordinary-double-point gap in every dimension (September 24, 2026).
• M2. OpenAI, The normalized-volume gap in dimension four (September 24, 2026).

Sources consulted on 12 October 2026 at repository revision fd4aeeb. Download the manuscript citations.