Views on level $$\ell $$ curves, K3 surfaces and Fano threefolds

An analogue of the Mukai map $$m_g: {\mathcal {P}}_g \rightarrow {\mathcal {M}}_g$$ m g : P g → M g is studied for the moduli $${\mathcal {R}}_{g, \ell }$$ R g , ℓ of genus g curves C with a level $$\ell $$ ℓ structure. Let $${\mathcal {P}}^{\perp }_{g, \ell }$$ P g , ℓ ⊥ be the moduli space of 4-tuples $$(S, {\mathcal {L}}, {\mathcal {E}}, C)$$ ( S , L , E , C ) so that $$(S, {\mathcal {L}})$$ ( S , L ) is a polarized K3 surface of genus g, $${\mathcal {E}}$$ E is orthogonal to $${\mathcal {L}}$$ L in $${{\,\mathrm{Pic}\,}}S$$ Pic S and defines a standard degree $$\ell $$ ℓ K3 cyclic cover of S, $$C \in \vert {\mathcal {L}} \vert $$ C ∈ | L | . We say that $$(S, {\mathcal {L}}, {\mathcal {E}})$$ ( S , L , E ) is a level $$\ell $$ ℓ K3 surface. These exist for $$\ell \le 8$$ ℓ ≤ 8 and their families are known. We define a level $$\ell $$ ℓ Mukai map $$r_{g, \ell }: {\mathcal {P}}^{\perp }_{g, \ell } \rightarrow {\mathcal {R}}_{g, \ell }$$ r g , ℓ : P g , ℓ ⊥ → R g , ℓ , induced by the assignment of $$(S, {\mathcal {L}}, {\mathcal {E}}, C)$$ ( S , L , E , C ) to $$ (C, {\mathcal {E}} \otimes {\mathcal {O}}_C)$$ ( C , E ⊗ O C ) . We investigate a curious possible analogy between $$m_g$$ m g and $$r_{g, \ell }$$ r g , ℓ , that is, the failure of the maximal rank of $$r_{g, \ell }$$ r g , ℓ for $$g = g_{\ell } \pm 1$$ g = g ℓ ± 1 , where $$g_{\ell }$$ g ℓ is the value of g such that $$\dim {\mathcal {P}}^{\perp }_{g, \ell } = \dim {\mathcal {R}}_{g,\ell }$$ dim P g , ℓ ⊥ = dim R g , ℓ . This is proven here for $$\ell = 3$$ ℓ = 3 . As a related open problem we discuss Fano threefolds whose hyperplane sections are level $$\ell $$ ℓ K3 surfaces and their classification.