Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 146 / 4

The symplectic neighborhood theorem says that a neighborhood of a compact symplectic submanifold is determined, up to symplectomorphism, by the restricted symplectic form and its symplectic normal bundle. More precisely, if $f:S_0\to S_1$ is a symplectomorphism and an isomorphism of symplectic normal bundles covers $f$, then that bundle isomorphism extends to a symplectomorphism between neighborhoods of $S_0$ and $S_1$.

Let $Q\subset {\mathbb{CP}}^2$ be a smooth complex conic. Its homology class is $2H$, so its self-intersection number isRescale the Fubini-Study form so that $Q$ and $C$ have the same symplectic area. Their normal bundles have opposite Euler numbers, $+4$ and $-4$, so the symplectic sum can be formed along $Q$ and $C$.

Concretely, remove tubular neighborhoods $\nu (C)$ and $\nu (Q)$ and glue the boundaries by a fiber-reversing bundle map. The symplectic neighborhood theorem supplies the standard models needed for the gluing, and the symplectic-sum construction supplies a symplectic form onThe second piece is a rational homology ball. Thus this operation replaces the neighborhood of the $-4$ sphere by that rational ball and is the symplectic rational blowdown of a minus-four sphere.

Let $F_0$ and $F_1$ be general homogeneous cubic forms with base locus $B$ of nine points. The incidence varietyis the blowup of ${\mathbb{CP}}^2$ at $B$. Projection onto the second factor is the elliptic fibration of the rational elliptic surfaceand its fibers are connected plane cubics. Over any base point $p\in B$, the exceptional curve $E_p=\{p\}\times {\mathbb{CP}}^1$ maps isomorphically to the base, so it is a holomorphic section.

Write $H$ for the pullback of a line and $E_1,\dots ,E_9$ for the exceptional classes. The fiber class and the canonical class of the rational elliptic surface areThe degree of $\pi {|}_C$ is $C\cdot F=d$. The Adjunction formula now yieldsTherefore the genus formula for a multisection of the rational elliptic surface is