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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.

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