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

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