On the Brill-Noether theory for K3 surfaces

• Autorzy: Maxim Leyenson
• Języki publikacji: EN

Abstrakty

EN

Let (S, H) be a polarized K3 surface. We define Brill-Noether filtration on moduli spaces of vector bundles on S. Assume that (c 1(E), H) > 0 for a sheaf E in the moduli space. We give a formula for the expected dimension of the Brill-Noether subschemes. Following the classical theory for curves, we give a notion of Brill-Noether generic K3 surfaces. Studying correspondences between moduli spaces of coherent sheaves of different ranks on S, we prove our main theorem: polarized K3 surface which is generic in the sense of moduli is also generic in the sense of Brill-Noether theory (here H is the positive generator of the Picard group of S). The harder part of the proof is proving the non-emptiness of the Brill-Noether loci. In the case of algebraic curves such a theorem, proved by Griffiths and Harris and, independently, by Lazarsfeld, is sometimes called the strong theorem of the Brill-Noether theory. We finish by considering a number of projective examples. In particular, we construct explicitly Brill-Noether special K3 surfaces of genus 5 and 6 and show the relation with the theory of Brill-Noether special curves.

Słowa kluczowe

EN

Algebraic surfaces; K3 surfaces; Vector bundles

Kategorie tematyczne

• 14J28: K 3 surfaces and Enriques surfaces
• 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli
• 14D20: Algebraic moduli problems, moduli of vector bundles

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 4
• Strony: 1486-1540

Daty

wydano

2012-08-01

online

2012-05-31

Twórcy

autor

Maxim Leyenson

• Faculty of Mathematics and Computer Science

Bibliografia

• [1] Arbarello E., Cornalba M., Griffiths P.A., Harris J., Geometry of Algebraic Curves. I, Grundlehren Math. Wiss., 267, Springer, New York, 1985
• [2] Brill A., Nöther M., Ueber die algebraischen Functionen und ihre Anwendugen in der Geometrie, Math. Ann., 1874, 7(2–3), 269–310 http://dx.doi.org/10.1007/BF02104804
• [3] Fulton W., Intersection Theory, Ergeb. Math. Grenzgeb., 2, Springer, Berlin, 1984
• [4] Green M., Lazarsfeld R., A simple proof of Petri’s theorem on canonical curves, In: Geometry Today, Rome, June 4–11, 1984, Progr. Math., 60, Birkhäuser, Boston, 129–142
• [5] Griffiths P., Harris J., On the variety of special linear systems on a general algebraic curve, Duke Math. J., 1980, 47(1), 233–272 http://dx.doi.org/10.1215/S0012-7094-80-04717-1
• [6] Griffiths P., Harris J., Principles of Algebraic Geometry, Wiley Classics Lib., John Wiley & Sons, New York, 1994
• [7] Grothendieck A., Eléments de Géométrie Algébrique. I. Le Langage des Schémas, Inst. Hautes Études Sci. Publ. Math., 4, Presses Universitaires de France, Paris, 1960
• [8] Grothendieck A., Eléments de Géométrie Algébrique. IV. Étude Locale des Schémas et des Morphismes de Schémas I, Inst. Hautes Études Sci. Publ. Math., 20, Presses Universitaires de France, Paris, 1964
• [9] Grothendieck A., Technique de descente et théorèmes d’existence en géométrie algébrique. V. Les schémas de Picard: theórèmes d’existence, In: Séminaire Bourbaki, 7(232), Société Mathématique de France, Paris, 1995, 143–161
• [10] He M., Espaces de modules de systèmes cohérents. I. Normalité, C. R. Acad. Sci. Paris Sér. I Math., 1997, 325(2), 183–188 http://dx.doi.org/10.1016/S0764-4442(97)84596-X
• [11] He M., Espaces de modules de systèmes cohérents. II. Nombres de Donaldson, C. R. Acad. Sci. Paris Sér. I Math., 1997, 325(3), 301–306 http://dx.doi.org/10.1016/S0764-4442(97)83960-2
• [12] He M., Espaces de modules de systèmes cohérents, Internat. J. Math., 1998, 9(5), 545–598 http://dx.doi.org/10.1142/S0129167X98000257
• [13] Huybrechts D., Lehn M., The Geometry of Moduli Spaces of Sheaves, 2nd ed., Cambridge Math. Lib., Cambridge University Press, Cambridge, 2010 http://dx.doi.org/10.1017/CBO9780511711985
• [14] Lazarsfeld R., Brill-Noether-Petri without degenerations, J. Differential Geom., 1986, 23(3), 299–307
• [15] Le Potier J., Systèmes Cohérents et Structures de Niveau, Astérisque, 214, Société Mathématique de France, Paris, 1993
• [16] Leyenson M., On the Brill-Noether theory for K3 surfaces II, preprint available at http://arxiv.org/abs/math/0602358
• [17] Maruyama M., Construction of moduli spaces of stable sheaves via Simpson’s idea, In: Moduli of Vector Bundles, Sanda, Kyoto, 1994, Lecture Notes in Pure and Appl. Math., 179, Dekker, New York, 1996, 147–187
• [18] Mukai S., Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Invent. Math., 1984, 77(1), 101–116 http://dx.doi.org/10.1007/BF01389137
• [19] Mukai S., On the moduli space of bundles on K3 surfaces, In: Vector Bundles on Algebraic Varieties, Bombay, January 9–16, 1984, Tata Inst. Fund. Res. Stud. Math., 11, Tata Institute of Fundamental Research, Clarendon Press, Oxford University Press, Bombay, New York, 1987, 341–413
• [20] Mumford D., Lectures on Curves on an Algebraic Surface, Ann. of Math. Stud., 59, Princeton University Press, Princeton, 1966
• [21] Saint-Donat B., Projective models of K − 3 surfaces, Amer. J. Math., 1974, 96, 602–639 http://dx.doi.org/10.2307/2373709
• [22] Tyurin A.N., On the intersection of quadrics, Uspekhi Matem. Nauk, 1975, 6(186)(30), 51–99 (in Russian)
• [23] Tyurin A.N., Cycles, curves and vector bundles on an algebraic surface, Duke Math. J., 1987, 54(1), 1–26 http://dx.doi.org/10.1215/S0012-7094-87-05402-0

Identyfikatory

DOI

10.2478/s11533-012-0069-8