Section 128

The complete classification

Part XVI · Family 3.8PDF page 431

Theorem 128.1. Let $X_U$ be a smooth member of Family 3.8, described by the basepoint-free pencil $U=\langle f,g\rangle$ of binary quartics. Then the following assertions hold.

$X_U$ is K-stable if and only if the degree-four morphism $\eta_U$ has no totally ramified point.
There is precisely one K-polystable member which is not K-stable. It is $\operatorname{Bl}_{\{ux^4+vy^4=0\}} (\mathbb P^1\times\mathbb P^2),$ where the second $\mathbb P^1$ is embedded as the smooth conic $[x:y]\mapsto[x^2:xy:y^2]$ .
Every other smooth member is strictly K-semistable. Its pencil has a unique totally ramified point and has the form $\langle x^4,y^4+ax^2y^2+bx^3y\rangle, \qquad (a,b)\ne(0,0).$ The isomorphism classes of these members are parametrized by $\mathbb P(2,3)$ , with coordinates $[a:b]$ of weights two and three.

In particular, $X_U$ is K-polystable if and only if $U$ is GIT-polystable. Equivalently, $X_U$ is K-polystable if and only if $(\mathbb P^1,\frac14R_U)$ is K-polystable. This confirms the conjecture in [3, Section 7.6].

Proof. Suppose first that every ramification index is at most three. Proposition 121.1 reduces the point estimates to the exceptional rulings over ramification points and the two kinds of joining lines. The singularities of the del Pezzo fibres are exactly those in Lemma 117.3. The estimates established above give the following complete list. $\begin{array}{l|c} \text{location of }p&\text{lower bound for }\delta_p(X_U)\\ \hline p\in E&12/11\\ p\notin E\cup\Sigma&24/23\\ A_1\text{ point of a fibre}&72/67\\ A_2\text{ point of a fibre}&384/383\\ \text{index-two ruling, away from }E\text{ and its singular point}&9/8\\ \text{index-three exceptional ruling, away from its singular point}&12/11\\ \text{joining line of type }(2,2),\text{ away from its endpoints}&72/67\\ \text{joining line of type }(3,1),\text{ away from its }A_2\text{ endpoint}&96/91. \end{array}$ In the index-two row, its point on $E$ may instead be assigned to the first row. The last row includes the endpoint over the simple root. Thus there is no omitted endpoint in this list. The respective local results are Theorems 122.1, 123.1, 124.1, 126.1, 127.1 and Lemmas 125.2 and 127.6.

All local thresholds include valuations with positive-dimensional centres through the point. It follows that $\delta(X_U)=\inf_{p\in X_U}\delta_p(X_U) \ge\frac{384}{383}>1.$ The valuative criterion proves K-stability.

If there is a totally ramified point, Proposition 119.1 proves K-semistability and gives exactly one K-polystable isomorphism class, namely $X_{U_0}$ . This member is not K-stable: the transformations $[x:y]\mapsto[\lambda x:y]$ , together with the corresponding target transformations, give a nontrivial torus in its automorphism group. The remaining members and their isomorphisms are precisely those of Lemma 118.2. Finally, Proposition 118.1 and Lemma 118.3 identify these same three strata for the pencils and the log curves. This proves all the assertions. ◻

Example 128.2. The pencil $\langle x^4+y^4,x^2y^2\rangle$ belongs to the K-stable locus. The pencil $\langle x^4,y^4\rangle$ gives the unique K-polystable member with a positive-dimensional automorphism group. The pencils $\langle x^4,y^4+x^2y^2\rangle, \qquad \langle x^4,y^4+x^3y\rangle$ give strictly K-semistable members representing the two coordinate points of $\mathbb P(2,3)$ . Their special degenerations to the polystable member were constructed above.