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

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

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**Zoom link:**** https://princeton.zoom.us/j/91248028438**

We conjecture that any connected component Q of the moduli space of triples (X, E=E_1+…+E_n, \Theta) 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 \Theta is an ample divisor not containing any 0-stratum of E, is unirational. More precisely, we conjecture that the compactification of Q inside the moduli space of Kollár-Shepherd-Barron-Alexeev stable pairs 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 variety 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. The reference is arXiv:2008.02299 joint with Hacking and Keel.