On the existence of curves with $A_k$-singularities on $K3$ surfaces

Galati, Concettina; Knutsen, Andreas Leopold

doi:10.4310/mrl.2014.v21.n5.a8

Cited by 15 publications

(37 citation statements)

References 20 publications

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“…2.4] or [GK,Thm 3.1] for a generalization of this result). Since, by construction, C does not contain subcurves lying in |hL R | for any 1 h m − 1 (this is like saying that the aforementioned singularities of C are nondisconnecting), one checks that, for a general deformation of [R, L R ] to a K3 surface [S, L], the curve C deforms to an irreducible, nodal curve in |mL| with a total of…”

Section: More Limits Of Nodal Curves On Reducible K3 Surfacesmentioning

confidence: 80%

Moduli of nodal curves on K3 surfaces

Ciliberto

Flamini

Galati

et al. 2017

Advances in Mathematics

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Abstract. We consider modular properties of nodal curves on general K3 surfaces. Let Kp be the moduli space of primitively polarized K3 surfaces (S, L) of genus p 3 and V p,m,δ → Kp be the universal Severi variety of δ-nodal irreducible curves in |mL| on (S, L) ∈ Kp. We find conditions on p, m, δ for the existence of an irreducible component V of V p,m,δ on which the moduli map ψ : V → Mg (with g = m 2 (p − 1) + 1 − δ) has generically maximal rank differential. Our results, which for any p leave only finitely many cases unsolved and are optimal for m 5 (except for very low values of p), are summarized in Theorem 1.1 in the introduction.

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Section: More Limits Of Nodal Curves On Reducible K3 Surfacesmentioning

confidence: 80%

Moduli of nodal curves on K3 surfaces

Ciliberto

Flamini

Galati

et al. 2017

Advances in Mathematics

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“…By standard deformation-theoretic arguments, we know that for a general (S, L) and any 0 ≤ ≤ L 2 2 + 1, a general point in the subspace of |L| parametrizing curves of geometric genus corresponds to a nodal curve. See also [5,6] for results pointing to the same direction. Back to Theorem 1.3, the bridge connecting c 2 (S) and Beauville-Voisin's 0-cycle o S is the locus Z of cusp points in the one-parameter family of elliptic curves (the precise definition will be given in Section 3).…”

Section: Introductionmentioning

confidence: 85%

Singularities of elliptic curves in $$K3$$ K 3 surfaces and the Beauville–Voisin zero-cycle

Lin

2014

Boll Unione Mat Ital

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Under some hypotheses on the singular type of the one-parameter family of elliptic curves in a primitively polarized K3 surface S determined by its polarization (which is expected to be true for a very general polarized K3 surface), we give a more geometric proof of the fact that the second Chern class of S is equal to 24 · o S in the Chow group of 0-cycles where o S is the Beauville-Voisin canonical 0-cycle.

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“…An easy local computation, using the fact that C is Cartier, shows that a node of C lying outside of Sing X automatically smooths as S deforms, see for instance [Ch], [Ga,§2] or [GK,Pf. of Lemma 3.4].…”

Section: Half K3 Surfaces and Half Nikulin Surfacesmentioning

confidence: 99%

Half Nikulin Surfaces and Moduli of Prym Curves

Knutsen¹,

Lelli-Chiesa²,

Verra³

2019

J. Inst. Math. Jussieu

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Let F N g be the moduli space of polarized Nikulin surfaces (Y, H) of genus g and let P N g be the moduli of triples (Y, H, C), with C ∈ |H| a smooth curve. We study the natural map χg : P N g → Rg, where Rg is the moduli space of Prym curves of genus g. We prove that it is generically injective on every irreducible component, with a few exceptions in low genus. This gives a complete picture of the map χg and confirms some striking analogies between it and the Mukai map mg : Pg → Mg for moduli of triples (Y, H, C), where (Y, H) is any genus g polarized K3 surface. The proof is by degeneration to boundary points of a partial compactification of F N g . These represent the union of two surfaces with four even nodes and effective anticanonical class, which we call half Nikulin surfaces. The use of this degeneration is new with respect to previous techniques.

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On the existence of curves with $A_k$-singularities on $K3$ surfaces

Cited by 15 publications

References 20 publications

Moduli of nodal curves on K3 surfaces

Moduli of nodal curves on K3 surfaces

Singularities of elliptic curves in $$K3$$ K 3 surfaces and the Beauville–Voisin zero-cycle

Half Nikulin Surfaces and Moduli of Prym Curves

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