Views on level ℓ curves, K3 surfaces and Fano threefolds

An analogue of the Mukai map mg: Pg→ Mg is studied for the moduli Rg,ℓ of genus g curves C with a level ℓ structure. Let Pg,ℓ⊥ be the moduli space of 4-tuples (S, L, E, C) so that (S, L) is a polarized K3 surface of genus g, E is orthogonal to L in PicS and defines a standard degree ℓ K3 cyclic cover of S, C∈ | L|. We say that (S, L, E) is a level ℓ K3 surface. These exist for ℓ≤ 8 and their families are known. We define a level ℓ Mukai map rg,ℓ:Pg,ℓ⊥→Rg,ℓ, induced by the assignment of (S, L, E, C) to (C, E⊗ OC). We investigate a curious possible analogy between mg and rg,ℓ, that is, the failure of the maximal rank of rg,ℓ for g= gℓ± 1 , where gℓ is the value of g such that dimPg,ℓ⊥=dimRg,ℓ. This is proven here for ℓ= 3. As a related open problem we discuss Fano threefolds whose hyperplane sections are level ℓ K3 surfaces and their classification.