Pencils and nets on curves arising from rank $1$ sheaves on K3 surfaces

  • Nils Henry Rasmussen

Abstract

Let $S$ be a K3 surface, $C$ a smooth curve on $S$ with $\mathcal{O} _S(C)$ ample, and $A$ a base-point free $g^2_d$ on $C$ of small degree. We use Lazarsfeld-Mukai bundles to prove that $A$ is cut out by the global sections of a rank $1$ torsion-free sheaf $\mathcal{G} $ on $S$. Furthermore, we show that $c_1(\mathcal{G} )$ with one exception is adapted to $\mathcal{O} _S(C)$ and satisfies $\operatorname{Cliff} (c_1(\mathcal{G} )_{|C})\leq \operatorname{Cliff} (A)$, thereby confirming a conjecture posed by Donagi and Morrison. We also show that the same methods can be used to give a simple proof of the conjecture in the $g^1_d$ case.

In the final section, we give an example of the mentioned exception where $h^0(C,c_1(\mathcal{G} )_{|C})$ is dependent on the curve $C$ in its linear system, thereby failing to be adapted to $\mathcal{O} _S(C)$.

References

Aprodu, M. and Farkas, G., Green's conjecture for curves on arbitrary $K3$ surfaces, Compos. Math. 147 (2011), no. 3, 839–851. https://doi.org/10.1112/S0010437X10005099

>

Barth, W. P., Hulek, K., Peters, C. A. M., and Van de Ven, A., Compact complex surfaces, second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 4, Springer-Verlag, Berlin, 2004. https://doi.org/10.1007/978-3-642-57739-0

>

Ciliberto, C. and Pareschi, G., Pencils of minimal degree on curves on a $K3$ surface, J. Reine Angew. Math. 460 (1995), 15–36.

Donagi, R. and Morrison, D. R., Linear systems on $K3$-sections, J. Differential Geom. 29 (1989), no. 1, 49–64.

Friedman, R., Algebraic surfaces and holomorphic vector bundles, Universitext, Springer-Verlag, New York, 1998. https://doi.org/10.1007/978-1-4612-1688-9

>

Green, M. and Lazarsfeld, R., Special divisors on curves on a $K3$ surface, Invent. Math. 89 (1987), no. 2, 357–370. https://doi.org/10.1007/BF01389083

>

Huybrechts, D. and Lehn, M., The geometry of moduli spaces of sheaves, second ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2010. https://doi.org/10.1017/CBO9780511711985

>

Knutsen, A. L., On two conjectures for curves on $K3$ surfaces, Internat. J. Math. 20 (2009), no. 12, 1547–1560. https://doi.org/10.1142/S0129167X09005881

>

Knutsen, A. L. and Lopez, A. F., A sharp vanishing theorem for line bundles on $K3$ or Enriques surfaces, Proc. Amer. Math. Soc. 135 (2007), no. 11, 3495–3498. https://doi.org/10.1090/S0002-9939-07-08968-X

>

Lazarsfeld, R., Brill-Noether-Petri without degenerations, J. Differential Geom. 23 (1986), no. 3, 299–307.

Lelli-Chiesa, M., Stability of rank-$3$ Lazarsfeld-Mukai bundles on $K3$ surfaces, Proc. Lond. Math. Soc. (3) 107 (2013), no. 2, 451–479. https://doi.org/10.1112/plms/pds087

>

Lelli-Chiesa, M., Generalized Lazarsfeld-Mukai bundles and a conjecture of Donagi and Morrison, Adv. Math. 268 (2015), 529–563. https://doi.org/10.1016/j.aim.2014.08.011

>

Tyurin, A. N., Cycles, curves and vector bundles on an algebraic surface, Duke Math. J. 54 (1987), no. 1, 1–26. https://doi.org/10.1215/S0012-7094-87-05402-0

>

Published
2018-04-08
How to Cite
Rasmussen, N. (2018). Pencils and nets on curves arising from rank $1$ sheaves on K3 surfaces. MATHEMATICA SCANDINAVICA, 122(2), 197-212. https://doi.org/10.7146/math.scand.a-97308
Section
Articles