Donagi–Morrison Conjecture on existence of surface lifts controlling special linear series on curves on K3 surfaces

Establish that for a K3 surface X with a base point free and big line bundle L and a smooth curve C in the linear system |L| of genus g, every base point free line bundle A on C of degree d ≤ g − 1 with negative Brill–Noether number admits a line bundle N on X such that (i) the complete linear system |A| is contained in the restriction of |N| to C; (ii) the Clifford index of the restricted line bundle N ⊗ O_C is at most the Clifford index of A; and (iii) the intersection number N · L is at most g − 1.

Background

The paper studies line bundles on smooth curves lying on K3 surfaces, focusing on the Brill–Noether–Petri property and the existence of certain lifts from the curve to the surface that control special linear series. Donagi and Morrison formulated a conjecture predicting that if a line bundle A on a curve C ⊂ X has negative Brill–Noether number, then there exists a line bundle N on the K3 surface X whose restriction to C contains |A| and whose Clifford index does not exceed that of A, with an additional intersection bound N * L ≤ g − 1.

The author proves a related existence result (Theorem 1.2) guaranteeing a line bundle N adapted to |L| with |A| ⊂ |N ⊗ O_C| and Clifford index inequality under explicit numerical conditions, but not necessarily the intersection bound (iii). The conjecture remains the overarching open problem driving this work.