Higher Rank Brill–noether Theory on Sections of K3 Surfaces
نویسندگان
چکیده
We discuss the role of K3 surfaces in the context of Mercat’s conjecture in higher rank Brill–Noether theory. Using liftings of Koszul classes, we show that Mercat’s conjecture in rank 2 fails for any number of sections and for any gonality stratum along a Noether– Lefschetz divisor inside the locus of curves lying on K3 surfaces. Then we show that Mercat’s conjecture in rank 3 fails even for curves lying on K3 surfaces with Picard number 1. Finally, we provide a detailed proof of Mercat’s conjecture in rank 2 for general curves of genus 11, and describe explicitly the action of the Fourier–Mukai involution on the moduli space of curves.
منابع مشابه
Brill-Noether theory on singular curves and vector bundles on K3 surfaces
Let C be a smooth curve. Let W r d be the Brill-Noether locus of line bundles of degree d and with r + 1 independent sections. W r d has a expected dimension ρ(r, d) = g − (r + 1)(g − d + r). If ρ(r, d) > 0 then Fulton and Lazarsfeld have proved that W r d is connected. We prove that this is still true if C is a singular irreducible curve lying on a regular surface S with −KS generated by globa...
متن کاملSpecial Determinants in Higher-rank Brill-noether Theory
Continuing our previous study of modified expected dimensions for rank-2 Brill-Noether loci with prescribed special determinants, we introduce a general framework which applies a priori for arbitrary rank, and use it to prove modified expected dimension bounds in several new cases, applying both to rank 2 and to higher rank. The main tool is the introduction of generalized alternating Grassmann...
متن کامل1 M ay 2 00 7 GROMOV - WITTEN THEORY AND NOETHER - LEFSCHETZ THEORY
Noether-Lefschetz divisors in the moduli of K3 surfaces are the loci corresponding to Picard rank at least 2. We relate the degrees of the Noether-Lefschetz divisors to the GromovWitten theory of 1-parameter families ofK3 surfaces. The reduced K3 theory and the Yau-Zaslow formula play an important role. We use results of Borcherds for O(2, 19) lattices and proven mirror transforms to determine ...
متن کاملVector Bundles and Brill–Noether Theory
After a quick review of the Picard variety and Brill–Noether theory, we generalize them to holomorphic rank-two vector bundles of canonical determinant over a compact Riemann surface. We propose several problems of Brill–Noether type for such bundles and announce some of our results concerning the Brill–Noether loci and Fano threefolds. For example, the locus of rank-two bundles of canonical de...
متن کاملVector Bundles with Sections
Classical Brill-Noether theory studies, for given g, r, d, the space of line bundles of degree d with r + 1 global sections on a curve of genus g. After reviewing the main results in this theory, and the role of degeneration techniques in proving them, we will discuss the situation for higher-rank vector bundles, where even the most basic questions remain wide open. Focusing on the case of rank...
متن کامل