Stability of rank two Ulrich bundles on projective $K3$ surfaces

  • Gianfranco Casnati
  • Federica Galluzzi

Abstract

Let $F\subseteq \mathbb{P}^{N}$ be a $K3$ surface of degree $2a$, where $a\ge 2$. In this paper we deal with Ulrich bundles on $F$ of rank $2$. We deal with their stability and we construct $K3$ surfaces endowed with families of non-special Ulrich bundles of rank $2$ for each $a\ge 2$.

References

Aprodu, M., Farkas, G., and Ortega, A., Minimal resolutions, Chow forms and Ulrich bundles on $K3$ surfaces, J. Reine Angew. Math. 730 (2017), 225–249. https://doi.org/10.1515/crelle-2014-0124

>

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

>

Casanellas, M. and Hartshorne, R., ACM bundles on cubic surfaces, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 3, 709–731. https://doi.org/10.4171/JEMS/265

>

Casanellas, M., Hartshorne, R., Geiss, F., and Schreyer, F.-O., Stable Ulrich bundles, Internat. J. Math. 23 (2012), no. 8, 1250083, 50 pp. https://doi.org/10.1142/S0129167X12500838

>

Casnati, G., Examples of smooth surfaces in $mathbb P^3$ which are Ulrich-wild, Bull. Korean Math. Soc. 54 (2017), no. 2, 667–677. https://doi.org/10.4134/BKMS.b160257

>

Casnati, G., On rank two aCM bundles, Comm. Algebra 45 (2017), no. 10, 4139–4157. https://doi.org/10.1080/00927872.2016.1222397

>

Casnati, G., Rank two aCM bundles on general determinantal quartic surfaces in $mathbb P^3$, Ann. Univ. Ferrara Sez. VII Sci. Mat. 63 (2017), no. 1, 51–73. https://doi.org/10.1007/s11565-016-0244-0

>

Casnati, G. and Notari, R., Examples of rank two aCM bundles on smooth quartic surfaces in $mathbb P^3$, Rend. Circ. Mat. Palermo (2) 66 (2017), no. 1, 19–41. https://doi.org/10.1007/s12215-016-0272-8

>

Coskun, E., Ulrich bundles on quartic surfaces with Picard number $1$, C. R. Math. Acad. Sci. Paris 351 (2013), no. 5-6, 221–224. https://doi.org/10.1016/j.crma.2013.04.005

>

Coskun, E., Kulkarni, R. S., and Mustopa, Y., Pfaffian quartic surfaces and representations of Clifford algebras, Doc. Math. 17 (2012), 1003–1028.

Coskun, E., Kulkarni, R. S., and Mustopa, Y., The geometry of Ulrich bundles on del Pezzo surfaces, J. Algebra 375 (2013), 280–301. https://doi.org/10.1016/j.jalgebra.2012.08.032

>

Eisenbud, D. and Herzog, J., The classification of homogeneous Cohen-Macaulay rings of finite representation type, Math. Ann. 280 (1988), no. 2, 347–352. https://doi.org/10.1007/BF01456058

>

Eisenbud, D., Schreyer, F.-O., and Weyman, J., Resultants and Chow forms via exterior syzygies, J. Amer. Math. Soc. 16 (2003), no. 3, 537–579. https://doi.org/10.1090/S0894-0347-03-00423-5

>

Faenzi, D., Rank $2$ arithmetically Cohen-Macaulay bundles on a nonsingular cubic surface, J. Algebra 319 (2008), no. 1, 143–186. https://doi.org/10.1016/j.jalgebra.2007.10.005

>

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 $k$th-order embeddings of $K3$ surfaces and Enriques surfaces, Manuscripta Math. 104 (2001), no. 2, 211–237. https://doi.org/10.1007/s002290170040

>

Knutsen, A. L., Smooth curves on projective $K3$ surfaces, Math. Scand. 90 (2002), no. 2, 215–231. https://doi.org/10.7146/math.scand.a-14371

>

Morrison, D. R., On $K3$ surfaces with large Picard number, Invent. Math. 75 (1984), no. 1, 105–121. https://doi.org/10.1007/BF01403093

>

Mukai, S., Symplectic structure of the moduli space of sheaves on an abelian or $K3$ surface, Invent. Math. 77 (1984), no. 1, 101–116. https://doi.org/10.1007/BF01389137

>

Nikulin, V. V., Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 111–177, English translation: Math. USSR-Izv. 14 (1980), 103–167. https://doi.org/10.1070/IM1980v014n01ABEH001060

>

Okonek, C., Schneider, M., and Spindler, H., Vector bundles on complex projective spaces, Progress in Mathematics, vol. 3, Birkhäuser, Boston, Mass., 1980.

Pons-Llopis, J. and Tonini, F., ACM bundles on del Pezzo surfaces, Matematiche (Catania) 64 (2009), no. 2, 177–211.

Saint-Donat, B., Projective models of $K$-$3$ surfaces, Amer. J. Math. 96 (1974), 602–639. https://doi.org/10.2307/2373709

>

Watanabe, K., ACM bundles on $K3$ surfaces of genus $2$, preprant arXiv:1407.1703 [math.AG], 2014.

Watanabe, K., The classification of ACM line bundles on quartic hypersurfaces in $mathbb P^3$, Geom. Dedicata 175 (2015), 347–354. https://doi.org/10.1007/s10711-014-9950-x

>

Published
2018-04-08
How to Cite
Casnati, G., & Galluzzi, F. (2018). Stability of rank two Ulrich bundles on projective $K3$ surfaces. MATHEMATICA SCANDINAVICA, 122(2), 239-256. https://doi.org/10.7146/math.scand.a-101999
Section
Articles