Secondary fan, theta functions and moduli of Calabi-Yau pairs

We conjecture that any connected component Q of the moduli space of triples (X,E=E₁+⋯+E_n,Θ) where X is a smooth projective variety, E is a normal crossing anti-canonical divisor with a 0-stratum, every E_i is smooth, and Θ is an ample divisor not containing any 0-stratum of E, is unirational. More precisely: note that Q has a natural embedding into the Kollár-Shepherd-Barron-Alexeev moduli space of stable pairs, we conjecture that its closure admits a finite cover by a complete toric variety. We construct the associated complete toric fan, generalizing the Gelfand-Kapranov-Zelevinski secondary fan for reflexive polytopes. Inspired by mirror symmetry, we speculate a synthetic construction of the universal family over this toric variety, as the Proj of a sheaf of graded algebras with a canonical basis, whose structure constants are given by counts of non-archimedean analytic disks. In the Fano case and under the assumption that the mirror contains a Zariski open torus, we construct the conjectural universal family, generalizing the families of Kapranov-Sturmfels-Zelevinski and Alexeev in the toric case. In the case of del Pezzo surfaces with an anti-canonical cycle of (−1)-curves, we prove the full conjecture.