Statement and method

Family 3.7: sextic fibres, quintic sections, and point extractions

Let $V\subset\mathbb P^2\times\mathbb P^2$ be a smooth divisor of bidegree $(1,1)$, and let $C\subset V$ be a smooth complete intersection of two members of $|\mathcal O_V(1,1)|$. We consider the blow-up $\pi:X\to V$ of $C$. These are the smooth members of Family 3.7. The elliptic curve $C$ and its two plane cubic embeddings control the singular fibres and the few points requiring extraction.

The proof uses Abban–Zhuang adjunction in three ways. The sextic del Pezzo pencil excludes all centres except those on the two lines through a fibre singularity. A smooth quintic del Pezzo section treats the open parts of these lines. Ordinary and weighted point extractions treat their remaining points. We compute the expected orders in separate lemmas, including the fixed parts of the surface refinements. The complete initial section algebras are established by Hilbert counts valid in every degree.

No Zariski decomposition on an extracted threefold is computed or assumed. The threefold decompositions below are on $X$ itself. On the point blow-ups the actual section spaces replace such a calculation. Their further refinements are controlled by complete binary spaces, surface resolutions, or explicit monomial limits.