In "Curves on Heisenberg invariant quartic surfaces in projective 3-space", Eklund showed that a general $(\mathbb{ Z}/2\mathbb{ Z})^{ 4}$-invariant quartic K3 surface contains at least $320$ conics. In this paper we analyse the field of definition of those conics as well as their Monodromy group. As a result, we prove that the moduli space $(\mathbb{ Z}/2\mathbb{ Z})^{ 4}$-invariant quartic K3 surface with a marked conic has $10$ irreducible components.
Categories: math.AG