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

TL;DR

This paper investigates the structure of the moduli space of smooth complete intersections of two quadrics in projective space, relating it to singular members and computing its Picard group for all dimensions n ≥ 3.

Contribution

It provides an alternative presentation of the moduli space and explicitly computes its Picard group across all relevant dimensions.

Findings

Computed the Picard group for all n ≥ 3

Established a relation between moduli space and singular members

Provided an alternative description of the moduli space

Abstract

We study the moduli space of smooth complete intersections of two quadrics in $\mathbb{P}^n$ by relating it to the geometry of the singular members of the corresponding pencils. Giving an alternative presentation for the moduli space of complete intersections, we compute the Picard group for all $n\geq 3$.