Half Nikulin Surfaces and Moduli of Prym Curves

Knutsen, Andreas Leopold; Lelli-Chiesa, Margherita; Verra, Alessandro

doi:10.1017/s1474748019000574

Cited by 6 publications

(9 citation statements)

References 42 publications

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“…These surfaces are known as (standard) Nikulin surfaces. Cases (1), ( 2), (3) are treated in [11,12], the remaining ones, (standard and non standard), in [19,20]. Notice that r g,2 is not of maximal rank for g = 6, 8.…”

Section: The Picture For =mentioning

confidence: 99%

“…For = 2 these families are known, [11,19,27]. The case (6, 2) is revisited here with emphasis on a singular quadratic complex of the Grassmannian G (2,5).…”

Section: Views On Fano Threefolds With Sections Of Level 2 Ormentioning

confidence: 99%

“…In particular Sing S = V ∩P 6 is an even set of 8 nodes, defining π|S withS = π −1 (S), cfr. [11,19,20]. For = 2 and [S, L, E] ∈ F ⊥ g , the surface S, or its model S, is known as a standard Nikulin surface of genus g. Therefore we can say that a general 3-dimensional linear section of T is a Fano threefold whose hyperplane sections are standard Nikulin surfaces of genus 6.…”

Section: 2mentioning

confidence: 99%

“…If = 2 we have dim P ⊥ g,2 = dim R g,2 for g = 7. Then r g,2 fails to be of maximal rank for g = 7 ± 1 and is birational for g = 7, [11,19,27]. The 'Fano varieties behind the scene' for g = 8 and g = 6 are addressed or revisited in Sect.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: The Picture For =mentioning

confidence: 99%

“…For = 2 these families are known, [11,19,27]. The case (6, 2) is revisited here with emphasis on a singular quadratic complex of the Grassmannian G (2,5).…”

Section: Views On Fano Threefolds With Sections Of Level 2 Ormentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Garbagnati

Verra

2021

Boll Unione Mat Ital

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“…Nikulin surfaces represent a rather special class of K3 surfaces, which have been studied in relation with various topics, including the theory of automorphisms [13,6], moduli spaces [12,7], the study of Prym curves [2] and of the birational geometry of their moduli spaces [4,5,10,16].…”

Section: Introductionmentioning

confidence: 99%

A note on Nikulin surfaces and their moduli spaces

Ramponi

2019

Math. Z.

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Double covers of curves on Nikulin surfaces

D’Evangelista,

Lelli-Chiesa

2024

Moduli Spaces and Vector Bundles—New Trends

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We survey basic results concerning Prym varieties, the Prym-Brill-Noether theory initiated by Welters, and Brill-Noether theory of general étale double covers of curves of genus g ≥ 2 g\geq 2 . We then specialize to curves on Nikulin surfaces and show that étale double covers of curves on Nikulin surfaces of standard type do not satisfy Welters’ Theorem. On the other hand, by specialization to curves on Nikulin surfaces of non-standard type, we prove that general double covers of curves ramified at b = 2 , 4 , 6 b=2,4,6 points are Brill-Noether general; the case b = 2 b=2 was already obtained by Bud [The birational geometry of R g , 2 ¯ \overline {\mathcal {R}_{g,2}} and Prym-canonical divisorial strata, math.AG/01718.v2] with different techniques.

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Half Nikulin Surfaces and Moduli of Prym Curves

Cited by 6 publications

References 42 publications

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

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

A note on Nikulin surfaces and their moduli spaces

Double covers of curves on Nikulin surfaces

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