Section 13

The limitations of the fibre curve flag

Part III · Family 2.9: conic bundlesPDF page 57

The curve flag $S\supset Z\supset P$ used in Lemma 12.3 need not give a strict bound at a point of $\ell$ . Proposition 12.4 resolves this issue by blowing up the point on $S$ . For comparison, we explain the limitation of the original curve flag and then give a separate proof using the exceptional plane of a threefold point blow-up. Double conic fibres require the different construction in Section 18.

Example 13.1 (A flag estimate that does not finish the proof). Suppose a smooth conic fibre $Z$ on a chosen smooth cubic surface is tangent to $\ell$ at $P$ , and $P\notin C_S$ . Then $r=2$ , $\mu=0$ , and the exact point term of Lemma 12.3 is $\mathcal T_P=\frac{101}{96}>1.$ The resulting lower bound for $\delta_P(X)$ is only $96/101$ . This does not prove instability: a lower bound below one says nothing of that kind. It shows precisely why this particular flag cannot complete a K-stability proof. Even a transverse intersection $r=1$ gives $\mathcal T_P=1$ when $\mu=3$ , and gives a value greater than one when $\mu>3$ .

For points above $p_0$ , the avoidance argument in Lemma 10.6 is unavailable. Every smooth cubic through such a point has its distinguished line through that point: in local coordinates $l=(x,y,z)$ and $q(p_0)=(0,0,1)$ , a point of $f_0$ has normal direction $[x:y]=[a:b]$ . The coefficients of cubics through it satisfy $b\lambda_0-a\lambda_1=0$ , exactly the condition that the line $l\parallel\lambda$ have that same normal direction. The line therefore cannot be chosen to avoid $p$ . Proposition 12.4 handles these points by blowing up $p$ on $S$ , retaining the tangent direction of $C_S=E|_S$ in the calculation on the exceptional curve. The exceptional-plane argument in Section 17 provides another proof.

At a double-line fibre, a general inverse image of a base line through its image need not be a smooth cubic surface. Locally a rank-one conic has equation $x^2+u q_1(y,z)+v q_2(y,z)+\text{higher base-order terms}=0.$ Smoothness of $X$ requires $q_1,q_2$ to have no common zero on the double line. On a base line $v=cu$ , singularities of the surface occur at zeros of $q_1+cq_2$ , so the smooth-surface flag used above is not available there. We therefore use the weighted curve extraction in Section 18.

For comparison, the two original curve flags give the following conditional reduction.

Corollary 13.2 (A conditional reduction from the curve flags). If $\delta_P(X)>1$ for every point $P\in T$ , then $X$ is K-stable.

Proof. For any prime divisor $F$ over $X$ , choose a point $P$ of its centre. If $P\notin T$ , Lemmas 12.3 and 12.6 give $\delta_P(X)>1$ . If $P\in T$ , use the hypothesis. In either case $A_X(F)/S_X(F)\ge\delta_P(X)>1$ . Apply the valuative criterion. ◻

The proof for reduced fibres now uses only the elementary fixed part on the original blow-up and Zariski decompositions on smooth cubic surfaces or their point blow-ups. The next sections give the alternative construction on the ordinary threefold blow-up of a point. Its induced series consists of leading homogeneous terms of anticanonical sections. We estimate its threshold in Section 17 using forced components on further blow-ups; the general higher-contact estimate does not require a complete threefold Zariski decomposition.