This video contains two seminars. This one starts at 1:03.
This is a report on work in progress with my student Cristian Rodriguez. The mirror of a ℚ-Fano 3-fold with b2 = 1 is a rigid K3 fibration over ℙ1 such that Hodge bundle is degree 1 and some power of the monodromy at infinity is maximally unipotent. Although prior work focused on the maximally unipotent case (without base change), perhaps a classification of such Picard-Fuchs equations is possible.
In the smooth case these fibrations were described explicitly by Przyjalkowski, and Doran-Harder-Novoseltsev-Thompson showed that they are given by etale covers of the (1-dimensional) moduli of rank 19 K3 surfaces. In the case of a single 1/2(1,1,1) singularity they are given by rigid rational curves on the (2-dimensional) moduli of rank 18 K3 surfaces, and examples suggest they are Teichmuller curves in A2 (via the Shioda-Inose correspondence relating rank 18 K3s and abelian surfaces), as studied by McMullen.
This video is part of the 3CinG annual meeting that took place in Warwick in September 2021.
