Mirror symmetry for lattice-polarised weak del Pezzo surfaces

Let $V$ be an $N$-polarised weak del Pezzo surface with smooth anticanonical divisor $C$, and let $\check{N}$ be the orthogonal complement of $N$ in $K_V^{\perp}$. A quasi del Pezzo homomorphism is a homomorphism

$$\phi\colon H_2(W,F_p;\mathbb{Z})\to H_1(F_p;\mathbb{Z}).$$

Here $W\to\Delta$ is an elliptic fibration over a closed disc, $p\in\partial\Delta$, and $F_p$ is the fibre over $p$.

Mirror symmetry for lattice-polarised weak del Pezzo surfaces. The mirror to $(V,C)$ is given by an elliptic fibration $W\to\Delta$ with fibre $F_p$ such that $\phi$ is a quasi del Pezzo homomorphism isomorphic to

$$\mathrm{K}_0^{\mathrm{num}}(\mathbf{D}(V))\to\mathrm{K}_0^{\mathrm{num}}(\mathbf{D}(C)),$$

and $W$ is $\check{N}$-polarised.

This conjecture is a lattice-polarised enhancement of the Landau–Ginzburg description of del Pezzo surfaces by open sets in rational elliptic surfaces. The source reformulates the conjecture of Doran and the second author in terms of elliptic fibrations and quasi del Pezzo homomorphisms; its resolution status is not specified here.