Abstract
In this paper we study examples of \({\mathbb P^r}\)-scrolls defined over primitively polarized K3 surfaces S of genus g, which arise from Brill–Noether theory of the general curve in the primitive linear system on S and from some results of Lazarsfeld. We show that such scrolls form an open dense subset of a component \({\mathcal H}\) of their Hilbert scheme; moreover, we study some properties of \({\mathcal H}\) (e.g. smoothness, dimensional computation, etc.) just in terms of \({\mathfrak F_g}\), the moduli space of such K3’s, and M v (S), the moduli space of semistable torsion-free sheaves of a given rank on S. One of the motivation of this analysis is to try to introducing the use of projective geometry and degeneration techniques in order to studying possible limits of semistable vector-bundles of any rank on a very general K3 as well as Brill–Noether theory of vector-bundles on suitable degenerations of projective curves. We conclude the paper by discussing some applications to the Hilbert schemes of geometrically ruled surfaces introduced and studied in Calabri et al. (Rend Lincei Mat Appl 17(2):95–123, 2006) and Calabri et al. (Rend Circ Mat Palermo 57(1):1–32, 2008).
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References
Altman A., Kleiman S.: Compactifying the Picard scheme. Adv. Math. 35, 50–112 (1980)
Aprodu M.: Green-Lazarsfeld gonality conjecture for a generic curve of odd genus. Int. Math. Res. Not. 63, 3409–3416 (2004)
Aprodu M., Voisin C.: Green-Lazarsfeld’s conjecture for generic curves of large gonality. C. R. Math. Acad. Sci. Paris 336(4), 335–339 (2003)
Arbarello E., Cornalba M.: Footnotes to a paper of Beniamino Segre. Math. Ann. 256, 341–362 (1981)
Arrondo, E., Pedreira, M., Sols, I.: On regular and stable ruled surfaces in \({\mathbb P^3}\). Algebraic curves and projective geometry (Trento, 1988), 1–15. With an appendix of R. Hernandez, 16–18. Lecture Notes in Math., vol. 1389, Springer, Berlin (1989)
Barth W., Hulek K., Peters C., Van de Ven A.: Compact Complex Surfaces, 2nd edn. Springer, Berlin (2004)
Beauville, A.: Fano threefolds and K3 surfaces. The Fano Conference, pp. 175–184. Univ. Torino, Turin (2004)
Beltrametti, M., Sommese, A.J.: The adjunction theory of complex projective varieties. de Gruyter Expositions in Mathematics, vol. 16. Walter de Gruyter & Co., Berlin (1995)
Calabri A., Ciliberto C., Flamini F., Miranda R.: Degenerations of scrolls to unions of planes. Rend. Lincei Mat. Appl. 17(2), 95–123 (2006)
Calabri A., Ciliberto C., Flamini F., Miranda R.: Non-special scrolls with general moduli. Rend. Circ. Mat. Palermo 57(1), 1–32 (2008)
Calabri A., Ciliberto C., Flamini F., Miranda R.: Brill–Noether theory and non-special scrolls with general moduli. Geom. Dedicata 139, 121–138 (2009)
Ciliberto C., Lopez A.F., Miranda R.: Projective degenerations of K3 surfaces, Gaussian maps and Fano threefolds. Invent. Math. 114(3), 641–667 (1993)
Cukierman F., Ulmer D.: Curves of genus ten on K3 surfaces. Compos. Math. 89, 81–90 (1993)
Eisenbud D., Harris J.: Irreducibility and monodromy of some families of linear series. Ann. scient. Éc. Norm. Sup. 20, 65–87 (1987)
Farkas G.: Brill–Noether loci and the gonality stratification of \({\mathcal{M}_g}\). J. Reine Angew. Math. 539, 185–200 (2001)
Farkas G., Popa M.: Effective divisors on M g , curves on K3 surfaces and the slope conjecture. J. Alg. Geom. 14, 241–267 (2005)
Friedman R.: Algebraic surfaces and holomorphic vector bundles. Universitext. Springer, New York (1998)
Fulton W., Lazarsfeld R.: On the connectedness of degeneracy loci and special divisors. Acta Math. 146, 271–283 (1981)
Green M., Lazarsfeld R.: Special divisors on curves on a K3 surface. Invent. Math. 89, 357–370 (1987)
Hartshorne R.: Algebraic Geometry (GTM No. 52). Springer, New York (1977)
Huybrechts, D., Lehn, M.: The geometry of moduli spaces of sheaves. Publication of the Max-Plank-Institut für Mathematik. Aspects in Mathematics, vol. 31. Vieweg, Bonn (1931)
Iitaka S.: Algebraic Geometry, Graduate Texts in Math., vol. 76. Springer, New York (1982)
Kosarew S., Okonek C.: Global moduli spaces and simple holomorphic bundles. Publ. RIMS, Kyoto Univ. 25, 1–19 (1989)
Lazarsfeld R.: Brill–Noether–Petri without degenerations. J. Differ. Geom. 23(3), 299–307 (1986)
Mayer A.: Families of K3 surfaces. Nagoya Math. J. 48, 1–17 (1972)
Mori, S., Mukai, S.: The uniruledness of the moduli space of curves of genus 11. In: Algebraic Geometry, Proc. Tokyo/Kyoto, pp. 334–353. Lecture Notes in Math., vol. 1016. Springer, Berlin (1983)
Mukai S.: Symplectic structure of the moduli space of sheaves on an abelian or K3 surface. Invent. Math. 77(1), 101–116 (1984)
Mukai, S.: On the moduli space of bundles on K3 surfaces. I. Vector bundles on algebraic varieties (Bombay, 1984), 341–413, Tata Inst. Fund. Res. Stud. Math., vol. 11. Tata Inst. Fund. Res., Bombay (1987)
Mukai, S.: Moduli of vector bundles on K3 surfaces and symplectic manifolds. Sugaku Expositions. 1(2), 139–174 (1988). Sūgaku 39(3), 216–235 (1987)
Mukai, S.: Curves, K3 surfaces and Fano 3-folds of genus ≤ 10. Algebraic geometry and commutative algebra, vol. I, pp. 357–377. Kinokuniya, Tokyo (1988)
Mukai, S.: Fano 3-folds. Complex projective geometry (Trieste–Bergen, 1989), pp. 255–263, London Math. Soc. Lecture Note Ser., 179, Cambridge Univ. Press, Cambridge (1992)
Ottaviani, G.: Varietà proiettive di codimensione piccola, Note Corso INDAM (1995)
Ottaviani G.: On 3-folds in \({\mathbb P^5}\) which are scrolls. Annali Scuola Normale Sup. Pisa, Ser. IV XIX(3), 451–471 (1992)
Pareschi G.: A proof of Lazarsfeld’s theorem on curves on K3 surfaces. J. Alg. Geom. 5, 195–200 (1995)
Saint-Donat B.: Projective models of K3 surfaces. Am. J. Math. 96, 602–639 (1974)
Sernesi, E.: Deformations of Algebraic Schemes. Grundlehren der Mathematischen Wissenschaften, vol. 334. Springer, Berlin (2006)
Voisin C.: Green’s generic syzygy conjecture for curves of even genus lying on a K3 surface. J. Eur. Math. Soc. 4(4), 363–404 (2002)
Voisin C.: Green’s canonical syzygy conjecture for generic curves of odd genus. Comp. Math. 141(5), 1163–1190 (2005)
Voisin, C.: Géométrie des espaces de modules de courbes et de surfaces K3 (d’après Gritsenko-Hulek-Sankaran, Farkas-Popa, Mukai, Verra, et al.), Séminaire Bourbaki. vol. 2006/2007. Astérisque 317, Exp. No. 981, 467–490 (2008)
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The author is a member of G.N.S.A.G.A. at I.N.d.A.M. “Francesco Severi”.
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Flamini, F. \({\mathbb P^r}\)-scrolls arising from Brill–Noether theory and K3-surfaces. manuscripta math. 132, 199–220 (2010). https://doi.org/10.1007/s00229-010-0343-7
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DOI: https://doi.org/10.1007/s00229-010-0343-7