The moduli of singular curves on K3 surfaces

Using lattice theory on special K 3 surfaces, calculations on moduli stacks of pointed curves and Voisin's proof of Green's Conjecture on syzygies of canonical curves, we prove the Prym-Green Conjecture on the naturality of the resolution of a general Prym-canonical curve of odd genus, as well as (many cases of) the Green-Lazarsfeld Secant Conjecture on syzygies of non-special line bundles on general curves.

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“…Indeed, this follows from the proof of [22,Lemma 1.3]. For precise details, we refer to [20,Lemma 5.2]). …”

Section: Lemma 33mentioning

confidence: 92%

The generic Green–Lazarsfeld Secant Conjecture

Farkas

Kemeny

2015

Invent. math.

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“…The primitively polarised case has been classically studied: for g 11, the map c prim g is birational onto its image if g = 12, whereas its generic fibre is irreducible of dimension 1 if g = 12 ([12, § 5.3] and [28]); for g 11, the map c prim g is dominant if g = 10 [27], and onto a hypersurface of M 10 if g = 10 [17], with irreducible general fibre in any case [13,12]. The non-primitively polarised cases have been studied in [9,23] where it is shown that, if g 11 then c g is generically finite in all but possibly finitely many cases.…”

Section: -Main Resultsmentioning

confidence: 99%

Wahl maps and extensions of canonical curves and K⁢3K3 surfaces

Ciliberto

Dedieu

Sernesi

2018

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Let C be a smooth projective curve (resp. (S, L) a polarized K3 surface) of genus g 11, non-tetragonal, considered in its canonical embedding in P g−1 (resp. in its embedding in |L| ∨ ∼ = P g ). We prove that C (resp. S) is a linear section of an arithmetically Gorenstein normal variety Y in P g+r , not a cone, with dim(Y ) = r+2 and ωY = OY (−r), if the cokernel of the Gauss-Wahl map of C (resp. H 1 (TS L ∨ )) has dimension larger or equal than r + 1 (resp. r). This relies on previous work of Wahl and Arbarello-Bruno-Sernesi. We provide various applications.A central theme of this text is the extendability problem: Given a projective (irreducible) variety X ⊂ P n , when does there exist a projective variety Y ⊂ P n+1 , not a cone, of which X is a hyperplane section? Given a positive integer r, an r-extension of X ⊂ P n is a variety Y ⊂ P n+r having X as a section by a linear space. The variety X is r-extendable if it has an r-extension that is not a cone, and extendable if it is at least 1-extendable. The following result provides a necessary condition for extendability.(0.1) Theorem (Lvovski [25]). Let X ⊂ P n be a smooth, projective, irreducible, non-degenerate variety, not a quadric. SetIf X is r-extendable and α(X) < n, then r α(X).This introduced the idea, explicit in Voisin's article [38], that the elements of (coker(Φ C )) ∨ (or rather of ker( T Φ C )) should be interpreted as ribbons, or infinitesimal surfaces, embedded in P g and extending C: see Section 4.The following statement is a first converse to Theorem (0.1), and a central element of the proofs of our Theorems (2.1) and (2.18).(0.2) Theorem (Wahl [43], Arbarello-Bruno-Sernesi [3]). Let C be a smooth curve of genus g 11, and Clifford index Cliff(C) > 2. Every ribbon v ∈ ker( T Φ C ) may be integrated to (i.e. is contained in) a surface S in P g having the canonical model of C as a hyperplane section.Note that if v = 0, then the surface S is not a cone as only the trivial ribbon may be integrated to a cone. Conversely, we observe that actually unicity holds in Theorem (0.2) (see (2.2.2) and Remark (4.8)): up to isomorphisms, given a ribbon v ∈ ker( T Φ C ), the surface S integrating it in P g is unique. For v = 0, this is the content of the aforementioned theorem of Wahl and Beauville-Mérindol, see (2.3).We prove a statement for K3 surfaces analogous to Theorem (0.2) (Theorem (2.17)).Theorem (0.2) provides a characterization of those curves having non-surjective Wahl map in the range g 11 and Cliff > 2. Wahl [42, p. 80] suggested to study the stratification of the moduli space of curves by the corank of the map Φ C : This is done in our Theorem (2.1) to the effect that, in the same range, the curves with cork(Φ C ) r are those which are r-extendable.We give various applications of our results, in particular to the smoothness of the fibres of the forgetful map which to a pair (S, C) associates the modulus of C, where S ⊂ P g is a K3 surface and C is a canonical curve hyperplane section of S (Theorem (2.6)). The same result is proven for the analog...

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“…I conclude the note (see Sect. 4) with some evidence toward the nonsurjectivity of the Wahl map w C itself, and comment on related work in [4].…”

Section: Theorem 1 (I) Let (Smentioning

confidence: 88%

“…(II) Related work In [4], the author extensively studies the properties of nodal curves on K 3 surfaces. Among several other results, he proves the nonsurjectivity of a marked Wahl map (different from the one introduced in here) for nodal curves on K 3 surfaces.…”

Section: β (β δ)mentioning

confidence: 99%

Erratum to: Modular properties of nodal curves on $$K3$$ K 3 surfaces

Halic

2015

Math. Z.

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The moduli of singular curves on K3 surfaces

Cited by 19 publications

References 62 publications

The generic Green–Lazarsfeld Secant Conjecture

The generic Green–Lazarsfeld Secant Conjecture

Wahl maps and extensions of canonical curves and K⁢3K3 surfaces

Erratum to: Modular properties of nodal curves on $$K3$$ K 3 surfaces

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