The generic Green–Lazarsfeld Secant Conjecture

Farkas, Gavril; Kemeny, Michael

doi:10.1007/s00222-015-0595-7

Cited by 22 publications

(52 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In odd genus, the conjecture has been established before for level 2 in [FK1] (using Nikulin surfaces) and for high level ℓ ≥ g+2 2 in [FK2] (using Barth-Verra surfaces). Theorem 0.1 therefore removes any restriction on the level ℓ.…”

mentioning

confidence: 82%

“…Theorem 0.1 therefore removes any restriction on the level ℓ. Apart from that, we feel that the rational elliptic surfaces used in this paper are substantially simpler objects than the K3 surfaces used in [FK1] and [FK2] and should have further applications to syzygy problems. The Prym-Green Conjecture in even genus, amounting to the single vanishing statement (2) K g 2 −2,1 (C, K C ⊗ τ ) = 0, (or equivalently, K g 2 −3,2 (C, K C ⊗ τ ) = 0) is still mysterious.…”

mentioning

confidence: 94%

“…Despite significant progress, the Secant Conjecture is not known for arbitrary L, but in [FK1] Theorem 1.7, we provided a sufficient condition for the vanishing to hold. Precisely, whenever…”

mentioning

confidence: 99%

See 2 more Smart Citations

The resolution of paracanonical curves of odd genus

Farkas

Kemeny

2018

Geom. Topol.

Self Cite

View full text Add to dashboard Cite

We prove the Prym-Green conjecture on minimal free resolutions of paracanonical curves of odd genus. The proof proceeds via curves lying on ruled surfaces over an elliptic curve. IntroductionThe study of torsion points on Jacobians of algebraic curves has a long history in algebraic geometry and number theory. On the one hand, torsion points of Jacobians have been used to rigidify moduli problems for curves, on the other hand, such a torsion point determines an unramified cyclic cover over the curve in question, which gives rise to a (generalized) Prym variety, see [BL] Chapter 12 for an introduction to this circle of ideas.Pairs [C, τ ], where C is a smooth curve of genus g ≥ 2 and τ ∈ Pic 0 (C) is a non-trivial torsion line bundle of order ℓ ≥ 2 form an irreducible moduli space R g,ℓ . One may view this moduli space as a higher genus analogue of the level ℓ modular curve X 1 (ℓ). There is a finite cover R g,ℓ → M g given by forgetting the ℓ-torsion point. Following ideas going back to Mumford, Tyurin and many others, linearizing the Abel-Prym embedding of the curve in its Prym variety leads to the study of the properties of [C, τ ] in terms of the projective geometry of the level ℓ paracanonical curve ϕ K C ⊗τ : C ֒→ P g−2 induced by the line bundle K C ⊗ τ . In practice, this amounts to a qualitative study of the equations and the syzygies of the paracanonical curve in question. For instance, in the case ℓ = 2, there is a close relationship between the study of these syzygies and the Prym map R g,2 → A g−1 to the moduli space of principally polarized abelian varieties of dimension g − 1, which has been exploited fruitfully for some time, see for instance [B1]. For higher level, the study of these syzygies has significant applications to the study of the birational geometry of R g,ℓ , see [CEFS].Denoting by Γ C (K C ⊗ τ ) := q≥0 H 0 C, (K C ⊗ τ ) ⊗q the homogeneous coordinate ring of the paracanonical curve, for integers p, q ≥ 0, let K p,q (C, K C ⊗ τ ) := Tor p Γ C (K C ⊗ τ ), C p+q be the Koszul cohomology group of p-th syzygies of weight q of the paracanonical curve and one denotes by b p,q := dim K p,q (C, K C ⊗ τ ) the corresponding Betti number.The Prym-Green Conjecture formulated in [CEFS] predicts that the minimal free resolution of the paracanonical curve corresponding to a general level ℓ curve [C, τ ] ∈ R g,ℓ of genus g ≥ 5 is natural, that is, in each diagonal of its Betti table, at most one entry is non-zero. The naturality of the resolution amounts to the vanishing statements b p,2 · b p+1,1 = 0, for all p. As explained in [CEFS], for odd genus g = 2n + 1 this is equivalent to the vanishing statements

show abstract

mentioning

confidence: 82%

mentioning

confidence: 94%

See 1 more Smart Citation

The resolution of paracanonical curves of odd genus

Farkas

Kemeny

2018

Geom. Topol.

Self Cite

View full text Add to dashboard Cite

show abstract

“…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.

View full text Add to dashboard Cite

show abstract

“…[26]). The Secant Conjecture holds for a general curve C of genus g and a general line bundle L of degree d on C.An elementary argument shows that if C is general then the general L ∈ Pic d (C) is (p+1)-very ample if and only if d ≥ g + 2p + 3.Using this inequality and the fact that if L is a globally generated, nonspecial, line bundle with b p,2 (C, L) = 0 then b p−1,2 (C, L(−x)) = 0 for a general x ∈ C, Theorem 4.3 reduces to finding a general curve C together with a non-special line bundle L ∈ Pic d (C) with b p,2 (C, L) = 0 in the following two cases(1) g = 2i + 1, d = 2p + 2i + 4, p ≥ i − 1 (2) g = 2i, d = 2p + 2i + 3, p ≥ i − 1.We construct such curves C and line bundles L using lattice polarized K3 surfaces.…”

mentioning

confidence: 99%