Brill-Noether-Petri without degenerations

Green's generic syzygy conjecture for curves of even genus lying on a K3 surface

“…As in Sect. 2, and following [11], it follows that there is a rank 2 vector bundle E on S with det E = L, c 2 (E) = k + 1, and h 0 (E) = k + 2. The zero set of a generic section of E is a generic member of a g 1 k+1 of a generic curve X ∈| L |.…”

“…Recall from [11] that if S is a K 3 surface endowed with a ample line bundle L such that L generates Pic S and L 2 = 2g − 2, the smooth members C ∈| L | are of genus g and generic in the sense of Brill-Noether, so that in particular they have the same Clifford index as a generic curve. Hence Conjecture 1 predicts that their syzygies vanish as stated in Conjecture 3.…”

Pencils of small degree on curves on unnodal Enriques surfaces

“…Let S be an Enriques surface, and let L be a line-bundle on , we can assume that for general (C, A) in W, |A| is base-point free. It thus makes sense to study the associated Lazarsfeld-Mukai vector bundles, F C,A and E C,A (see [Laz86]). …”

New examples of curves with a one-dimensional family of pencils of minimal degree

“…Following the work of Lazarsfeld and Tyurin [10,13], for any base-point free line-bundle A on C ∈ |mH|, there is defined a rank-2 vector bundle E C,A by…”

“…In 1986, it was proved by Lazarsfeld [10] that if the linear system |L| on a K3 surface S doesn't contain non-reduced or reducible curves, then dim W r d (C) = ρ(g, r, d) for general C ∈ |L|. Knutsen [9] proved that the only cases of exceptional curves (i.e., curves C satisfying Cliff(C) < gon(C)−2) on K3 surfaces are the Donagi-Morrison example [3, (2.2)] and the generalised ELMS example (a generalisation of [4,Theorem 4.3] presented in Knutsen's article, see "Generalised ELMS examples").…”