The ordinary-double-point gap in every dimension

1. Preserve volume and obtain an isolated Gorenstein cone

Stable degeneration and its K-polystable refinement preserve the minimizing valuation's value. Lifting generators and upper semicontinuity in the flat degeneration also give $\operatorname{edim}(X,x)\le\operatorname{edim}(C,o)$, so the final cone is singular. If a puncture point $P$ were singular, an étale product of an $(n-1)$-dimensional transverse slice with a smooth curve would give $d(P,C)\le p_{n-1}$ by induction and Lemma 2.1. Lower semicontinuity gives $d(o,C)\le d(P,C)$, contradicting $p_{n-1}<p_n$. Applying the finite degree formula to the canonical index-$q$ cover gives $d(o,C)\le1/q$; since $p_n>1/2$, the index is one.

Proposition 2.5 · pp. 10–11 [XZ25, Theorem 1.2, p. 2] [LWX21, Theorem 1.2, p. 2] [BL21, Theorem 33(1), p. 23] [XZ21, Theorem 1.3, p. 1] [LX19, Theorem 1.3(1), p. 3] [OpenAI26b, Theorem 1.1, p. 1]

2. Small stabilizers force a free grading and high-index classification

Normalize $A_C(\xi)=n$ and approximate by primitive positive integral gradings $\zeta_i=(r_i/n)\xi+\epsilon_i$ with absolute error $\epsilon_i\to0$. The smooth orbifold $B_i=[C^\circ/\mathbb G_m]$ has a faithful weight-one ample line bundle $L_i$ and $-K_{B_i}=r_iL_i$. If $m_{\max}\le r/N$, the Li–Zhou curve theorem supplies a minimal-degree map $\mathbb P(1,m)\to B$. Unless the deformation character count directly forces freeness, its evaluation map is finite and surjective.

The symmetric product of infinitesimal translations over all sheets gives a nonzero rank-$b$ tensor of integral weight $-bm$. A Bochner argument for the Ricci-flat cone metric bounds its Reeb weight by $\tau\ge-b$; estimates in fixed ring generators control the character error independently of $b$. Thus the integer $nm-r$ is at most $o(1)$. Rounding and faithfulness of the tangent action force all stabilizers to be trivial and $r=N$ or $N+1$. High-index classification of the resulting smooth polarized Fano variety identifies its section ring with the quadric ring or a polynomial ring. Singularity of $C$ excludes the latter.

Lemmas 2.6–2.7 · pp. 11–12 Theorem 3.1; proof · pp. 13–19 [LZ25, Proposition 2.6, p. 6] [AOV11, Theorem 4.3, p. 17] [CS19, Theorem 1.1, print p. 1340 / PDF p. 2] [ADK08, Theorems 1.1 and 1.3, pp. 1–2]

3. Large stabilizers lead to an inequality for the supremum

For $m=m_{\max}\ge r/N$, record original degree and transverse Taylor initial in a semigroup $\Gamma$. Moment estimates from near-minimality and an exponential expectation inequality put $c_0=(r,1,\ldots,1)$ inside its real cone $K$. Adjoint lifting then yields $\Gamma=K\cap\Lambda$ and $\operatorname{int}K\cap\Lambda=c_0+\Gamma$. The Theorem 5.3 section explains this exact integral strengthening.

Two filtrations produce a central ring $G=D[v]$ with Hilbert series $(1-z)^{-n}$. A point with a different stabilizer $\mu_h$, $h\ge2$, gives a singular klt slice $Y$ of the same dimension $n$. The share $1/h$ of each character in leading colength yields $d(o,C)\le(p_n+M_n)/2$, where $M_n$ is the supremum of all singular $n$-dimensional klt densities. Applying this to germs approaching $M_n$ proves $M_n\le p_n$. The original $m$ and the new $h$ belong to different actions.

Theorem 4.8 · p. 29 Theorem 5.3 · pp. 34–36 Proposition 6.4; Corollary 6.6 · pp. 41–43 Theorem 7.7 · p. 48

4. Classify equality on the cone and return to the germ

When $d(o,C)=p_n$, every inequality in the slice estimate is an equality. Thus $h=2$ and, after normalizing $u(\lambda_Y)=1$, one has $A_Y(u)=N$ and $\operatorname{vol}(u)=2$. The restricted valuation $w$ satisfies $A_C(w)=2N$ and $\operatorname{vol}(w)=2^{-N}$ and computes the minimum; uniqueness gives $w=(2N/n)\operatorname{wt}_\xi$. If the distinguished section $v\in S_m$ had $w(v)>2$, a positive leading-order loss in the low-filtration quotient would make the volume estimate strict. Hence $w(v)\le2$. Absolute approximation gives $Nm-r\le o(1)$, so integrality returns the grading to $Nm\le r$, the small-stabilizer branch. Theorem 7.11 therefore identifies $C$ as the ODP cone.

The original singular germ has embedding dimension at most $n+1$, hence exactly $n+1$, and is a hypersurface. Lemma 8.1 supplies the weighted bound

where $F=0$ is a local equation, $w_i>0$ are rational weights, and $d_F$ is its least weight. Weights all equal to one exclude multiplicity $q>2$. If $q=2$ but the quadratic part omits a direction, giving that direction weight $1-\varepsilon$ yields the strict bound $2(n-1-\varepsilon)^n/(1-\varepsilon)<2(n-1)^n$. Equality forces a nondegenerate quadratic part, and the holomorphic Morse lemma gives an analytic ODP. An ODP degeneration alone would not establish this last assertion.

Lemmas 7.8–7.10; Theorem 7.11 · pp. 49–50 [XZ21, Theorem 1.1; Corollary 1.2, p. 1] Lemma 8.1; Proposition 8.2 · pp. 51–52 [EM04, Theorem 1.1, p. 1] Proof of Theorem 1.1 · p. 52

5. Sharpness requires the infimum at an ODP

For the converse, Lemma 2.3 uses uniqueness to make a quadric cone's minimizer invariant under cone scaling and $\mathrm{SO}(n+1,\mathbb C)$. Irreducibility of each graded harmonic-polynomial representation forces it to be a positive multiple of the ordinary degree valuation. That valuation has discrepancy $n-1$ and volume two, so the infimum over all valuations is $2(n-1)^n$. This is stronger than testing only the ordinary blowup to obtain an upper bound. Completion invariance transfers the value to an analytic ODP, establishing sharpness and the reverse equality implication.

Lemma 2.3 · pp. 8–9 [Gra15, Example 251; Theorem 257, pp. 56–57] [Liu22, Lemma 2.15, p. 7]

1. Establish interiority numerically

Rescale original degree $j$ to $D_0=Nj/r$, so the cone is the epigraph $D_0\ge f(x)$ with lattice density $s=r/(Nm)\le1$. Put $H=\int_{\mathbb R_{\ge0}^N}e^{-f(x)}dx$. The volume lower bound gives $H\ge2/s$; near-minimality of valuations mixing degree with Taylor order gives a second moment inequality. Failure of interiority would give $f(\mathbf1)\ge N$. A supporting linear form then allows the uniform exponential estimate of Theorem 4.4, whose constant $C_*=(3/4)e^{4/3}-e/3<2$ contradicts $H\ge2$. The asymptotic semigroup count used here does not assume saturation.

Lemmas 4.2–4.3 · pp. 22–23 Theorem 4.4; proof · pp. 24–28 Theorem 4.8 · p. 29

2. Prove adjoint lifting independently of interiority

Lemma 5.2 says that $(j,\alpha)\in\Lambda$ and $(j,\alpha)+c_0\in\operatorname{int}K$ imply $(j,\alpha)\in\Gamma$. Choose positive integral weights agreeing with finitely many known Taylor initials and construct the weighted blowup as a smooth stack. The canonical shift cancels the exceptional cost, leaving the nef and big class $(j+r-j_0)\pi^*L$ in (5.6). Howald's monomial multiplier-ideal formula places the prescribed exceptional monomial in the required ideal; orbifold vanishing in Lemma 5.1 lifts the section. Its coarse-space reduction uses Fujino's theorem for a klt pair and an integral $\mathbb Q$-Cartier Weil divisor. Retaining characters on the smooth atlases is essential for descent.

Lemmas 5.1–5.2 · pp. 30–34 [How01, Main Theorem, p. 2] [Fuj23, Theorem 3.2.4, print p. 31 / PDF p. 41]

3. Height-one supports give finite generation and the canonical module

If a supporting ray at a differentiability point were irrational, or its primitive integral form had height at least two on $c_0$, lattice approximation would produce $q\notin K$ with $q+c_0\in\operatorname{int}K$. Adjoint lifting forbids this. A small ball around $c_0$ bounds the norms of height-one supports, so only finitely many integral supports occur. This proves rational polyhedrality.

For $q\in K\cap\Lambda$, translation by $c_0$ reaches the interior, hence lifting gives $q\in\Gamma$. Conversely, each primitive facet takes an integer value at least one on an interior lattice point, so subtracting $c_0$ remains in $K$. These are the two set equalities; finite generation follows. The interior-translation formula makes the normal semigroup ring's canonical module principal, supplying the Gorenstein property and degree $(r,N)$ needed in Lemma 6.1. Saturation alone would not imply Gorensteinness.

Theorem 5.3 · pp. 34–36 [Sta78, Theorem 6.7; normal-monoid reformulation, print pp. 81–82 / PDF pp. 25–26] Lemma 6.1 · pp. 37–39

1. Construct a singular klt slice of the same dimension

The first filtration $T$ is transverse Taylor order: $\bar S=\operatorname{gr}_T S$ and $D=\bar S/(\bar v)$. The second assigns $W=a+T(e_i)$ to a basis $v^ae_i$, giving $\operatorname{gr}_W S=D[v]$. Reducedness of $D$ uses the volume hypothesis and $N\ge4$. If every puncture stabilizer were trivial, Hilbert series $(1-z)^{-N}$ would force $D$ to be a polynomial ring generated in degree one, and ultimately $C$ to be smooth. Thus a stabilizer $\mu_h$, $h\ge2$, exists.

A transverse slice $Y$ in the $W$-Rees total space through that point has a function $\lambda_Y$ of character $-1$. Comparison of canonical forms gives, for a quasi-monomial valuation $u$ of $Y$, $\ell=u(\lambda_Y)$ and its restriction $w$ to $C$,

If $Y$ were not klt, tests with $A_Y(u)/\ell$ tending to a nonpositive value would give $\widehat{\operatorname{vol}}(o,C)\le N^n$, contradicting the lower bound $2N^n$. Smoothness of $Y$ would produce a nonzero $\mu_h$-fixed point in the other deformation's general fibre $\mathbb A^N$, also a contradiction. Establishing singularity is necessary before using $M_n$.

Proposition 6.4; Corollary 6.6 · pp. 41–43 Lemmas 7.1–7.3 · pp. 44–46

2. Restriction to one character improves the coefficient

Uniqueness makes the minimizer $u_*$ on $Y$ invariant under $\mu_h$. With $\sigma=\operatorname{vol}(u_*)$, injections between character spaces induced by powers of $\lambda_Y$ show that each character carries $1/h$ of the leading colength. Splitting the valuation quotient by the filtration, the second part injects into one character quotient, rather than the full quotient. Thus

Combining this with the discrepancy estimate and Hölder's inequality, and putting $A'=A_Y(u_*)$, yields

No same-dimensional ODP bound has been applied to $Y$.

Lemma 7.4; Proposition 7.5 · p. 47 Lemma 7.6 · p. 48

3. The supremum need not be attained

If $M_n>p_n$, choose germs with $d_i>p_n$ and $d_i\to M_n$. On each corresponding cone an infinite small-stabilizer subsequence contradicts Theorem 3.1 and the ODP minimum. A sufficiently far large-stabilizer grading therefore exists on each fixed cone. Passing to the limit in $d_i\le(p_n+M_n)/2$ contradicts $M_n>p_n$. Approximation constants may depend on the cone; no uniform rate over all germs and no attained supremum is required.

Theorem 7.7 · p. 48

1. Obtain the stricter bound for singular Fano varieties

Equivalence of K-semistability and Ding semistability and the local-to-global inequality for every centered valuation give

Apply Theorem 1.1 at a singular closed point to obtain $2((n^2-1)/n)^n$. This is strictly below $2n^n$; it does not establish sharpness or classify equality at the smaller value.

[Liu18, Theorem 21, p. 14] [Liu18, Theorem 13(2), p. 9] Proof of Corollary 1.2 · p. 53

2. Reduce equality at $2n^n$ to the smooth classification

For smooth $V\not\simeq\mathbb P^n$, apply Li–Miao. The strict singular bound excludes singular equality cases, so equality at $2n^n$ forces a smooth quadric or $\mathbb P^1\times\mathbb P^{n-1}$. The Li–Miao input concerns smooth Fano manifolds and is not applied directly to singular $V$.

[LM26, Theorem 1.1, p. 2]

3. Compute equality on the two examples

The manuscript uses the homogeneous and product Kähler–Einstein metrics to ensure K-polystability of these examples. For hyperplane classes $H,H_1,H_2$,

When $n=2$, $Q^2\simeq\mathbb P^1\times\mathbb P^1$, so the descriptions are not distinct examples. The proof of the general metric-to-K-polystability theorem is outside this overview's checking scope.

Proof of Corollary 1.2 · p. 53 [Ber16, Theorem 1.1, p. 2]