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

Views on level ℓ curves, K3 surfaces and Fano threefolds / A. Garbagnati, A. Verra. - In: BOLLETTINO DELLA UNIONE MATEMATICA ITALIANA. - ISSN 1972-6724. - (2021). [Epub ahead of print] [10.1007/s40574-021-00306-4]

Views on level ℓ curves, K3 surfaces and Fano threefolds

A. Garbagnati
;
2021

Abstract

An analogue of the Mukai map mg: Pg→ Mg is studied for the moduli Rg,ℓ of genus g curves C with a level ℓ structure. Let Pg,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 PicS 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 rg,ℓ:Pg,ℓ⊥→Rg,ℓ, induced by the assignment of (S, L, E, C) to (C, E⊗ OC). We investigate a curious possible analogy between mg and rg,ℓ, that is, the failure of the maximal rank of rg,ℓ for g= gℓ± 1 , where gℓ is the value of g such that dimPg,ℓ⊥=dimRg,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.
Settore MAT/03 - Geometria
21-set-2021
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/872407
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