Section 57

A second proof for Family 2.16: the CM polarization

Part IX · Family 2.16: conics on intersections of two quadricsPDF page 207

We give an independent proof of K-stability for every smooth member of Family 2.16. It uses the parameter space proposed in [28, Section 6] and the relative reconstruction proved in [34, Appendix A]. The new steps are the calculation of the CM line and a Hilbert–Mumford inequality for every smooth pair. CM-degree minimization then compares two fillings of the same punctured family. In particular, we do not assume the conjectural identification of the entire K-moduli space with a GIT quotient. None of the local threshold estimates in the preceding sections is used.

Theorem 57.1. Let $V\subset\mathbb P^5$ be a smooth intersection of two quadrics and let $C\subset V$ be a smooth conic. Then $\mathop{\mathrm{Bl}}_CV$ is K-stable.

Put $E=\mathbb C^6$ and use the convention that $\mathbb P(E)$ parametrizes lines. On $\operatorname{Gr}(3,E)$ let $\mathcal S\subset E\otimes\mathcal O$ be the tautological rank-three subbundle. Set $\mathcal H=\mathbb P_{\operatorname{Gr}(3,E)}(\operatorname{Sym}^2\mathcal S^*), \qquad \mathcal G=\operatorname{Gr}(2,\operatorname{Sym}^2E^*).$ A point of $\mathcal H$ is a plane $\Pi=\mathbb P(S)$ together with a nonzero quadratic equation on $\Pi$ , considered up to a scalar. Thus $\mathcal H$ parametrizes plane conics, including reducible and double conics. Let $\mathcal P$ denote the tautological rank-two bundle on $\mathcal G$ . We use additive notation for line bundles and write $\zeta=\mathcal O_{\mathcal H}(1),\qquad \xi=\det\mathcal S^*,\qquad \eta=\det\mathcal P^*.$ Their pullbacks will have the same names. Define $W=\{(C,P)\in\mathcal H\times\mathcal G: P\subset H^0(\mathbb P^5,\mathcal I_C(2))\}.$ Since $h^0(\mathcal I_C(2))=16$ for every plane conic, $W\to\mathcal H$ is a relative $\operatorname{Gr}(2,16)$ . In particular, $W$ is smooth, projective and irreducible. Its points need not all define Fano threefolds. The locus $W^\mathrm{sm}$ on which $V=V(P)$ and $C$ are smooth is a nonempty open subset. The group $\operatorname{SL}(E)$ acts on all these spaces. Passing to $\operatorname{PGL}(E)$ does not change stability; a suitable positive power of any linearization below descends through the finite centre.

Definition 57.2. For a flat family $f:\mathcal X\to T$ of Fano threefolds with relatively ample $\mathbb Q$ -Cartier anticanonical divisor, we normalize the CM $\mathbb Q$ -line bundle by $c_1(\lambda_{\mathrm{CM},f})=-f_*c_1(-K_{\mathcal X/T})^4.$ Equivalently, this is the corresponding Deligne pairing with its sign reversed. It is functorial under base change. Multiplying this normalization by a positive rational number changes none of the stability or degree comparisons below.

Lemma 57.3. The CM line of the universal blow-up family is the restriction of $\label{f216:eq:git-cm} \mathcal L=6\zeta+20\xi+66\eta\in\operatorname{Pic}(W).$ This equality holds, with the natural linearizations, both on the smooth Fano locus and for the relative blow-up families furnished by [34, Proposition A.13].

Proof. We calculate in the ambient blow-up, so the centre need not be a regular embedding in $V$ . A plane conic is a regular embedding of codimension four in $\mathbb P^5$ , even when it is a double line. Let $\mathcal C$ be the universal conic and let $h=c_1(\mathcal O_{\mathbb P^5}(1))$ . On $\mathbb P(\mathcal S)$ its class is $2h+\zeta$ . For the projection $p:\mathcal C\to W$ we have $\label{f216:eq:git-conic-push} p_*h=2,\qquad p_*h^2=2\xi+\zeta.$ Indeed, on $\mathbb P(\mathcal S)$ the pushforwards of $h^2$ and $h^3$ are $1$ and $-c_1(\mathcal S)=\xi$ .

Write $N=N_{\mathcal C/(\mathbb P^5\times W)}$ . The normal bundle of the relative plane is $\mathcal O(1)\otimes(E/\mathcal S)$ and the normal bundle of the conic in this plane is $\mathcal O(2)\otimes\mathcal O_{\mathcal H}(1)$ . The normal bundle sequence gives, modulo terms pulled back from base classes of codimension at least two, $\begin{split} c_1(N)&=5h+\xi+\zeta,\\ c_2(N)&=9h^2+4h\xi+3h\zeta,\\ c_1(N)^2-c_2(N)&=16h^2+6h\xi+7h\zeta. \end{split}$ The omitted terms make no contribution to the divisor classes being computed. Therefore $\label{f216:eq:git-normal-push} \begin{split} p_*(h\,c_1(N))&=12\xi+7\zeta,\\ p_*(c_1(N)^2-c_2(N))&=44\xi+30\zeta. \end{split}$

Let $b:Z=\mathop{\mathrm{Bl}}_{\mathcal C}(\mathbb P^5\times W)\to\mathbb P^5\times W$ and denote its exceptional divisor by $F$ . Put $z=2h-F$ and let $q:Z\to W$ be the projection. The blow-up formula for a regular embedding of codimension four is $b_*F^j=0\ (1\le j\le3),\quad b_*F^4=-[\mathcal C],\quad b_*F^5=-i_*c_1(N),\quad b_*F^6=-i_*(c_1(N)^2-c_2(N)),$ where $i$ is the inclusion of $\mathcal C$ . On a fibre, $\deg N=10$ , so $\label{f216:eq:git-ambient-push} \begin{split} q_*z^5&=32-10\deg C+\deg N=22,\\ q_*z^6&=-60p_*h^2+12p_*(h\,c_1(N)) -p_*(c_1(N)^2-c_2(N))\\ &=-20\xi-6\zeta. \end{split}$ The strict transform $\mathcal X$ of $V$ is the zero locus of the two transformed quadrics. Its class and relative anticanonical class are $[\mathcal X]=z^2+z\eta+c_2(\mathcal P^*),\qquad -K_{\mathcal X/W}=(z-\eta)|_{\mathcal X}.$ Here adjunction uses $K_{Z/W}=-6h+3F$ . These identities apply on the locus where the strict transform is the blow-up and is a relative complete intersection. This includes the smooth families and the reconstructed families by [34, Lemma A.12 and Proposition A.13]. All the ambient formulas remain valid for singular plane conics, since their embeddings in the ambient projective bundle are regular.

Keeping the terms contributing a divisor on the base gives $f_*(-K_{\mathcal X/W})^4=q_*(z^6-3\eta z^5) =-6\zeta-20\xi-66\eta.$ This proves (57.1). To keep track of the line bundle, rather than only its numerical class, use the Koszul complex of the two transformed quadrics on $Z$ . Its degree-four determinant-of-cohomology coefficient after twisting by $m(z-\eta)$ , with the negative sign and normalization above, defines a rational line bundle on all of $W$ . On every flat complete-intersection locus it is the CM line just calculated. Thus this is a functorial extension even at parameter points where we do not have a Fano family. The Picard group of the projective and Grassmann bundles is freely generated by the three displayed classes, and $\operatorname{SL}(E)$ has no characters. The computed class therefore identifies this extension and its linearization with $\mathcal L$ . On a nontrivial rank-six projective bundle the calculation is made after taking frames; changing frames gives exactly this linearization. ◻

Example 57.4. Fix the plane and a pencil of conics in it. On this curve of $\mathcal H$ , $\xi=0$ and $\deg\zeta=1$ , so $p_*h^2$ has degree one. This is also the number of conics in the pencil through the point cut out by two general hyperplanes. For the ambient blow-up, the second formula in (57.4) consequently has degree $-6$ . For the full family the pencil of quadrics contributes the additional term $-66\deg\eta$ . This separates the contribution of the centre from that of the threefold.

Lemma 57.5. The line bundle $\mathcal L$ is ample on $W$ .

Proof. The bundle $\operatorname{Sym}^2\mathcal S\otimes(\det\mathcal S^*)^2 \simeq\operatorname{Sym}^2(\bigwedge^2\mathcal S^*)$ is globally generated. Its pullback surjects onto $\mathcal O_{\mathcal H}(\zeta+2\xi)$ . Thus $\zeta+2\xi$ is globally generated and relatively ample over $\operatorname{Gr}(3,E)$ . Adding the pullback of the ample Plücker class shows that $\zeta+3\xi$ is ample on $\mathcal H$ . Now $\mathcal L=6(\zeta+3\xi)+2\xi+66\eta$ is the restriction of an ample line bundle on $\mathcal H\times\mathcal G$ : the first summand is ample on the first factor, $\xi$ is nef, and $\eta$ is ample on the second factor. ◻