Curves on Brill-Noether special K3 surfaces
Richard Haburcak
TL;DR
This work establishes that for polarized K3 surfaces of genus $g\le 19$, the Brill–Noether generality of the surface $(S,H)$ is equivalent to the Brill–Noether generality of smooth curves in $|H|$. The authors deploy Lazarsfeld–Mukai bundles, generalized LM bundles, and Donagi–Morrison lifts to lift BN data from curves to the surface, using stable quotients and lattice-polarization analysis to produce BN markings on $\mathrm{Pic}(S)$. They prove that in genus $14\le g\le 19$ the Donagi–Morrison lifting framework suffices to certify Mukai’s conjecture, with a bounded strong DM version ensuring control over the necessary lifts. Consequently, they provide a unified, algebraic-geometry-based proof extending Mukai’s model-based results to genera beyond 12, 14–19 in particular, and clarify the interplay between BN theory on curves and BN theory on K3 surfaces. The results have implications for the structure of NL divisors in $\mathcal{K}_g$ and for understanding how lattice-polarizations govern Brill–Noether phenomena on $K3$ surfaces.
Abstract
Mukai showed that projective models of Brill-Noether general polarized K3 surfaces of genus $6-10$ and $12$ are obtained as linear sections of projective homogeneous varieties, and that their hyperplane sections are Brill-Noether general curves. In general, the question, raised by Knutsen, and attributed to Mukai, of whether the Brill-Noether generality of any polarized K3 surface $(S,H)$ is equivalent to the Brill-Noether generality of smooth curves $C$ in the linear system $|H|$, is still open. Using Lazarsfeld-Mukai bundle techniques, we answer this question in the affirmative for polarized K3 surfaces of genus $\leq 19$, which provides a new and unified proof even in the genera where Mukai models exist.