Geometry and the local estimates

Family 2.10: quartic del Pezzo fibrations

Let $V\subset\mathbb P^5$ be a smooth complete intersection of two quadrics, and let $C$ be its smooth intersection with a linear subspace $\mathbb P^3$. Put $X=\mathop{\mathrm{Bl}}_C(V)$, and denote the blowup by $\pi\colon X\to V$ and its exceptional divisor by $E$. These are the smooth Fano threefolds in Family 2.10 of the Mori–Mukai classification; see [3, §4.3]. The pencil of hyperplanes containing $C$ gives a quartic del Pezzo fibration $f\colon X\to\mathbb P^1$. If $H=\pi^*\mathcal O_V(1)$ and $F$ is the fibre class, then $$F=H-E,\qquad L=-K_X=H+F=2H-E,\qquad L^3=16.$$ Every fibre is normal and has at worst Du Val singularities. This geometric fact, which does not require a group action, is recalled in [23, proof of Lemma 4.3]. Each fibre is smooth along $C$: otherwise its intersection with the second hyperplane defining $C$ would not be smooth.

A direct application of adjunction to a fibre does not settle all cases. Some singular quartic del Pezzo surfaces have lines on whose smooth loci the local stability threshold is less than one. We call them special lines in this paper. Their fibre types are $$A_2+A_1,\quad A_2+2A_1,\quad A_3+A_1,\quad A_3+2A_1,\quad A_4,\quad D_5.$$ They are specified by equations in Lemma 22.17. The following estimates treat these lines and the singular points for which the nodal-section argument does not apply.

Here $\delta_p$ takes the infimum over divisors whose centres contain $p$, not only those whose centres equal $p$. These local bounds therefore exclude the corresponding curve centres as well. The Family 2.10 case of Theorem 1.1 is proved in Section 21, using Theorem 19.1, which is proved in Section 22.

How we avoid Zariski decompositions on the extracted threefold. We identify the valuation filtration on every $H^0(X,mL)$ with a weight filtration of an explicit bigraded ring. Hilbert-weight calculations then give the expected orders, and further refinements are computed using polynomial spaces or monomial limits on the exceptional surfaces. In the ordinary point extraction along a special line, the seven original generators are insufficient. The additional section $r=\alpha au+bc$ has order three although its displayed summands have order two. Keeping this section and proving saturation gives $S_X(G)=65/32$ and a refined threshold at least $32/27$. Merely assigning weights to the original generators would not prove either assertion. We do use a Zariski decomposition on an auxiliary surface in the nodal argument, but never on the threefold obtained by extracting the point.

In the $D_4$ standard monomials below, the factor $z^\eta$, $\eta=0,1$, records the two residue classes for passage to the coarse exceptional plane. In type $D_5$, the corresponding factor is $v^\eta$. Both classes are retained with their actual fractional zero divisors. Their fixed orders are bounded per section and hence contribute $O(1/m)$ to a normalized expected order. This is the convention of Proposition 2.7.