Section 23

Geometry and the family theorem

Part V · Family 2.11: blow-ups of cubic threefolds along linesPDF page 104

Family 2.11: blow-ups of cubic threefolds along lines

Let $V\subset\mathbb P^4$ be a smooth cubic threefold, let $\ell\subset V$ be a line, and let $\pi:X=\mathop{\mathrm{Bl}}_\ell V\to V$ . Put $H=\pi^*\mathcal O_V(1)$ , write $E$ for the exceptional divisor, and set $B=H-E$ and $L=-K_X=H+B$ . These are the members of Family 2.11. Projection from $\ell$ resolves to a conic bundle $f:X\to\mathbb P^2$ , with $B=f^*\mathcal O_{\mathbb P^2}(1)$ .

A line is of the first kind if $N_{\ell/V}\cong\mathcal O_\ell^{\oplus2}$ , and of the second kind if $N_{\ell/V}\cong\mathcal O_\ell(1)\oplus\mathcal O_\ell(-1)$ . These are the only possibilities, as we check below. We prove the following results.

Theorem 23.1. Every smooth member $X=\mathop{\mathrm{Bl}}_\ell V$ of Family 2.11 is K-stable. Every point $p$ on a nonreduced conic fibre satisfies $\delta_p(X)\ge144/127$ . For a second-kind line, every point on the special reduced fibre has local threshold at least $36/35$ , and every smooth tangency point described in Lemma 29.1 has local threshold exactly $216/167$ .

The two normal-bundle types require different arguments. For first-kind lines, refinement along $E$ gives a strict beta inequality for every centre contained in $E$ . For second-kind lines, we first treat the special reduced fibre and exclude all curve centres. The remaining point centres are tangencies on smooth conic fibres; a weighted point extraction treats them. We prove each local calculation separately and assemble the exclusions in Section 30. In particular, we do not use openness of K-stability to pass to second-kind limits.

The general-member result in [3, Section 5.3] uses a particular cubic with a dihedral group action. The list in [23] still includes Family 2.11 among the families without a complete classification of K-polystable smooth members. Our first-kind argument uses no automorphisms of $X$ . The torus actions used later act on auxiliary exceptional surfaces and preserve their differents.

Lemma 23.2 (The plane-bundle model). Let $Y=\mathop{\mathrm{Bl}}_\ell\mathbb P^4$ . Its Cox coordinates can be written as $u,v,e,x_0,x_1,x_2$ , of degrees $\deg u=\deg v=H,\qquad \deg e=E=H-B,\qquad \deg x_i=B.$ The strict transform $X\subset Y$ has equation $\label{f211:eq:equation} G=u^2l_0(x)+uvl_1(x)+v^2l_2(x) +e\bigl(uq_0(x)+vq_1(x)\bigr)+e^2c(x)=0,$ where $l_i,q_i,c$ have degrees one, two and three in $x$ , respectively. This equation has class $2H+B$ . The map $f$ is a flat family of nonzero plane conics. One has $H^3=3,\quad H^2B=3,\quad HB^2=2,\quad B^3=0, \qquad L^3=18.$

Proof. Choose coordinates $[u:v:z_0:z_1:z_2]$ with $\ell=(z_0=z_1=z_2=0)$ . Substitute $z_i=ex_i$ in the cubic equation and divide by $e$ . This gives (23.1). The map $Y\to\mathbb P^2_x$ is the projective plane bundle with relative coordinates $[u:v:e]$ ; the degree convention above accounts for the twisting of $e$ .

No plane lies on a smooth cubic threefold: if a plane did lie on it, the two normal derivatives of its equation would be quadrics on that plane and would have a common zero, where the threefold would be singular. Consequently no conic equation in the plane bundle is identically zero. A relative quadratic Cartier divisor whose equation is nonzero on every plane fibre is flat over the base. This proves the assertion about $f$ .

The blow-up intersection formulas give $H^2E=0$ , $HE^2=-1$ and $E^3=-\deg N_{\ell/V}=0$ , since $\deg N_{\ell/V}=(-K_V)\ell-2=0$ . Substitute $B=H-E$ . ◻

Lemma 23.3 (The two kinds of lines). In the two cases above, $E$ is respectively $\mathbb P^1\times\mathbb P^1$ and the Hirzebruch surface $\mathbb F_2$ . If $g$ denotes a fibre of $E\to\ell$ , then $\begin{array}{c|cc} E&H|_E&-E|_E\\\hline \mathbb P^1\times\mathbb P^1&g&s\\ \mathbb F_2&g&s+g \end{array}$ where $s$ is a section with $s^2=0$ in the first row and the negative section, with $s^2=-2$ , in the second. The three normal derivatives of the cubic along $\ell$ span all binary quadrics in the first case and a base point free pencil of binary quadrics in the second.

Proof. The normal sequence is $0\longrightarrow N_{\ell/V}\longrightarrow\mathcal O_{\mathbb P^1}(1)^{\oplus3} \longrightarrow\mathcal O_{\mathbb P^1}(3)\longrightarrow0.$ Its last map is given by the three binary quadrics that are the normal derivatives along $\ell$ . They have no common zero. Write $N_{\ell/V}=\mathcal O(a)\oplus\mathcal O(-a)$ , with $a\ge0$ . Its inclusion in $\mathcal O(1)^{\oplus3}$ gives $a\le1$ . Moreover, a copy of $\mathcal O(1)$ in its kernel is equivalent to a constant linear relation among the three quadrics. Their span cannot have dimension one, since a binary quadratic has a zero. This proves the last assertion and the dichotomy. Projectivizing the normal bundle gives the asserted ruled surfaces. The tautological normal bundle of $E$ and $H|_E=\pi|_E^*\mathcal O_\ell(1)$ give the displayed classes; they also satisfy $(-E|_E)^2=0$ and $g(-E|_E)=1$ . ◻

Lemma 23.4 (Smooth cubic sections through a reduced fibre). If $\ell$ is of the first kind, every reduced conic fibre is contained in a surface $S\in|B|$ whose image under $\pi$ is a smooth cubic surface and for which $\pi|_S$ is an isomorphism. If $\ell$ is of the second kind, the same holds except possibly for one base point $y_0\in\mathbb P^2$ . On such a surface put $A=H|_S$ and $c=E|_S$ . Then $A=-K_S,\quad c\text{ is a line},\quad F=B|_S=A-c, \quad A^2=3,\quad Ac=1,\quad c^2=-1.$

Proof. Hyperplanes containing $\ell$ are parametrized by a plane. Those singular on $V$ at a specified point of $\ell$ are specified by the normal derivatives there. Their locus in the parameter plane is a smooth conic in the first-kind case and a line in the second-kind case, by Lemma 23.3. Hyperplanes containing the plane of a fixed conic fibre form a line in that parameter plane. It meets the former locus in finitely many points, unless in the second-kind case the two lines coincide. The latter event determines one point $y_0$ of the conic-bundle base.

A general inverse image of a base line through a reduced fibre is smooth on $X$ . Bertini applies away from the fibre; at its smooth points the conic bundle is smooth. At the node of a reducible fibre, smoothness of the total space supplies a nonzero base derivative of the conic equation, and a general base line retains that derivative. Avoid also the finitely many hyperplanes singular along $\ell$ . The image surface is then smooth both along and away from $\ell$ . Blowing up the Cartier divisor $\ell$ on that surface is an isomorphism. Adjunction and restriction give the formulas. ◻