Looking geometry from the moduli spaces of CICYs

I will discuss some examples of multiple LCSLs in the moduli spaces of Calabi–Yau complete intersections described by Gorenstein cones. It turns out that most examples are related to the linear duality due to Kuznetsov and also the conjecture by Batyrev and Nill. These examples are generalizations of the complete intersections of five $(1,1)$ divisors in ${\mathbb P}^4\times {\mathbb P}^4$, which I studied with Hiromichi Takagi finding interesting relation to the geometry of three dimensional Reye congruences.