Section 7

Geometry and the two remaining local cases

Part II · Family 2.5: cubic-surface fibrationsPDF page 33

Family 2.5: cubic-surface fibrations

Let $V\subset\mathbb P^4$ be a smooth cubic threefold, let $\Pi\subset\mathbb P^4$ be a plane such that $C=V\cap\Pi$ is a smooth cubic curve, and let $\pi:X=\mathop{\mathrm{Bl}}_C(V)\to V$ be the blow-up. Write $E_0$ for its exceptional divisor and $H=\pi^*\mathcal O_V(1)$ . Define $L=-K_X=2H-E_0$ and $A=H-E_0$ . The pencil of hyperplanes through $\Pi$ defines a morphism $\varphi:X\to\mathbb P^1$ whose fibres have class $A$ . These are exactly the smooth threefolds in Family 2.5. A fibre is a normal cubic surface, identified with the corresponding hyperplane section of $V$ .

The following result on K-stability of smooth threefolds in Family 2.5 was shown in [8].

Lemma 7.1 ([8, Section 2]). If $X$ is not K-stable, there exist a prime divisor $F$ over $X$ and a point $P$ of its centre such that $A_X(F)\le S_X(F)$ , $P\notin E_0$ , and the fibre through $P$ has a singularity of type $D_5$ or $E_6$ at $P$ .

Proof. By the valuative criterion, there exists a prime divisor $F$ over $X$ with $\beta_X(F)\le0$ . Then [8, Lemma 2.2 and preceding paragraph] show that the centre of $F$ on $X$ is not a surface and does not meet $E_0$ . Let $P$ be a point of the centre of $F$ on $X$ . Then the fiber of $\pi$ over $\pi(P)$ has a Du Val singularity of type $D_4$ , $D_5$ , or $E_6$ at $P$ by [8, paragraph before Lemma 2.8] and [8, Lemma 2.10]. Finally, [8, paragraph before Lemmas 2.11] and [8, Lemmas 2.11, 2.12, and 2.13] show that $P$ is not a $D_4$ singularity (note that on [8, page 19] there is a typo: $S(W^A_{\bullet,\bullet};L_1)<\frac43$ should instead read $S(W^A_{\bullet,\bullet};L_1)>\frac34$ ). ◻

In this section, we prove the following two results; together with Lemma 7.1, these will establish K-stability of $X$ .

Theorem 7.2. Suppose that $P\in X\setminus E_0$ is a singular point of type $D_5$ of a fibre of $\varphi$ . Then $\delta_P(X)\ge96/95$ .

Theorem 7.3. Suppose that $P\in X\setminus E_0$ is a singular point of type $E_6$ of a fibre of $\varphi$ . Then $\delta_P(X)\ge16/15$ .

The main computational point deserves emphasis. We do not compute a Zariski decomposition of $\sigma^*L-uG$ on the weighted blow-up $\sigma:\widehat X\to X$ . Instead, we identify the actual filtration of $H^0(X,mL)$ by $\mathop{\mathrm{ord}}_G$ , including all cancellations, and compute its associated graded ring. Its weighted Hilbert function gives $S_X(G)$ . A second, principal filtration gives a series on a rational curve; its complete polynomial intervals give all the point terms in the Abban–Zhuang inequality. An alpha estimate deals with centres away from that curve.

This should be distinguished from a claim that Zariski decompositions play no role anywhere in the argument. We retain the published reduction, whose proofs use them. For example, [8, Lemma 2.1] decomposes $L-uA$ on the original threefold. What is avoided here is any new threefold Zariski-decomposition calculation in the two missing local cases. We neither determine the birational chambers of $\widehat X$ nor assume that it is a Mori dream space. The replacement is an exact description of the refined section spaces, not merely a count of unconstrained local monomials.

7.1 Why no Zariski decomposition is required

In geometric volume calculations one may write $\sigma^*L-uG=P(u)+N(u)$ on suitable birational models, compute the volume from the positive part, and restrict positive and fixed parts to obtain the refinement. In dimension three, making this explicit can require the chamber structure and changes of model. This is a method of computing the input to Proposition 2.7; it is not a hypothesis of that proposition.

Here the computation proceeds entirely in the section ring. First, a global bigrading records exactly which polynomials represent anticanonical sections. Second, a weight filtration is proved to coincide with $\mathop{\mathrm{ord}}_G$ . Third, the tangent curve is cut out by a prime homogeneous element, and saturation identifies its powers with the second geometric filtration. The resulting finite-dimensional spaces determine both expected vanishing and fixed factors on the curve. The alpha estimate then controls all centres outside the curve. We use the general refinement theorem, without a Mori dream space assumption on $\widehat X$ .

Thus no Zariski decomposition on the weighted threefold blow-up is computed or needed in either local proof. There is still a global issue to solve: a local weighted polynomial need not come from an anticanonical section. The degree inequalities and the injectivity and saturation arguments below are precisely what solve that issue. Omitting them would invalidate the replacement of volumes by monomial counting.