Twists of Mukai bundles and the geometry of the level $3$ modular variety over $\overline {\mathcal {M}}_{8}$

Bruns, Gregor

doi:10.1090/tran/7239

Cited by 2 publications

(4 citation statements)

References 18 publications

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“…The existence of these families implies that r g,ℓ has not maximal rank. They correspond to the peculiar values (58) (g, ℓ) = (6, 3), (6, 2), (8, 2), (4,3).…”

Section: 2mentioning

confidence: 98%

“…We recall that R g,3 is of general type for g ≥ 12 and of Kodaira dimension ≥ 19 for g = 11, [7]. Bruns proved in [4]…”

Section: 2mentioning

confidence: 99%

“…In section 5 we summarize the question for each ℓ. Let g ℓ be the unique value of g such that dim P ⊥ g,ℓ = dim R g,ℓ , for l = 2, 3, 4, 5, 6, 7, 8 we respectively have: (4) g ℓ = 7 , 5 , 4 , 3 , 2 , 2 , 2.…”

Section: Introductionmentioning

confidence: 99%

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

Garbagnati,

Verra

2021

Preprint

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An analogue of the Mukai map mg : Pg → Mg is studied for the moduli R g,ℓ of genus g curves C with a level ℓ structure. Let P ⊥ g,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 Pic S 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 r g,ℓ : P ⊥ g,ℓ → R g,ℓ , induced by the assignment of (S, L, E, C) to (C, E ⊗ O C ). We investigate a curious possible analogy between mg and r g,ℓ , that is, the failure of the maximal rank of r g,ℓ for g = g ℓ ± 1, where g ℓ is the value of g such that dim P ⊥ g,ℓ = dim R g,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.

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“…The existence of these families implies that r g,ℓ has not maximal rank. They correspond to the peculiar values (58) (g, ℓ) = (6, 3), (6, 2), (8, 2), (4,3).…”

Section: 2mentioning

confidence: 98%

“…We recall that R g,3 is of general type for g ≥ 12 and of Kodaira dimension ≥ 19 for g = 11, [7]. Bruns proved in [4]…”

Section: 2mentioning

confidence: 99%

See 1 more Smart Citation

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

Garbagnati,

Verra

2021

Preprint

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show abstract

“…We recall that R g,3 is of general type for g ≥ 12 and of Kodaira dimension ≥ 19 for g = 11, [7]. Bruns proved in [6] that R 8,3 is of general type. The cases g = 6, 7, 9, 10 and partially g = 11 are open.…”

Section: The Picture For =mentioning

confidence: 99%

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

Garbagnati

Verra

2021

Boll Unione Mat Ital

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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.

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Twists of Mukai bundles and the geometry of the level $3$ modular variety over $\overline {\mathcal {M}}_{8}$

Cited by 2 publications

References 18 publications

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

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

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

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