The Picard group of the moduli of smooth complete intersections of two quadrics

We study the moduli space of smooth complete intersection of two quadrics by relating it to the geometry of the singular members of the corresponding pencil. We give a new description for this parameter space by using the fact that two quadrics can be simultaneously diagonalized. Using this description we can compute the Picard group, which always happens to be cyclic. For example, we show that the Picard group of the moduli stack of smooth degree 4 Del Pezzo surfaces is ${\mathbb Z}/4{\mathbb Z}$. This is a joint work with Shamil Asgarli.