Minimal resolutions, Chow forms and Ulrich bundles on K3 surfaces

Marian Aprodu 1 , Gavril Farkas 2 ,  and Angela Ortega 3
  • 1 Institute of Mathematics “Simion Stoilow”, Romanian Academy, Calea Griviţei 21, Sector 1, 010702, Bucharest, Romania
  • 2 Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany
  • 3 Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany
Marian Aprodu
  • Corresponding author
  • Institute of Mathematics “Simion Stoilow”, Romanian Academy, Calea Griviţei 21, Sector 1, 010702, Bucharest, Romania, Faculty of Mathematics and Computer Science, University of Bucharest, 14 Academiei Str., 010014 Bucharest, Romania
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar
, Gavril Farkas and Angela Ortega

Abstract

The Minimal Resolution Conjecture (MRC) for points on a projective variety X predicts that the Betti numbers of general sets of points in X are as small as the geometry (Hilbert function) of X allows. To a large extent, we settle this conjecture for a curve C with general moduli. We then proceed to find a full solution to the Ideal Generation Conjecture for curves with general moduli. In a different direction, we prove that K3 surfaces admit Ulrich bundles of every rank. We apply this to describe a pfaffian equation for the Chow form of a K3 surface.

  • [1]

    M. Aprodu and G. Farkas, The Green conjecture for smooth curves lying on arbitrary K 3 {K3} surfaces, Compos. Math. 147 (2011), 839–851.

    • Crossref
    • Export Citation
  • [2]

    A. Beauville, Determinantal hypersurfaces, Michigan Math. J. 48 (2000), 39–64.

    • Crossref
    • Export Citation
  • [3]

    A. Beauville, Some stable vector bundles with reducible theta divisors, Manuscripta Math. 110 (2003), 343–349.

    • Crossref
    • Export Citation
  • [4]

    J. Brennan, J. Herzog and B. Ulrich, Maximally generated Cohen–Macaulay modules, Math. Scand. 61 (1987), 181–203.

    • Crossref
    • Export Citation
  • [5]

    M. Casanellas and R. Hartshorne, ACM bundles on cubic surfaces, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 3, 709–731.

  • [6]

    M. Casanellas and R. Hartshorne, Stable Ulrich bundles, Internat. J. Math. 23 (2012), Article ID 1250083.

  • [7]

    A. Chiodo, D. Eisenbud, G. Farkas and F.-O. Schreyer, Syzygies of torsion bundles and the geometry of the level {\ell} modular varieties over ¯ g {\overline{\mathcal{M}}_{g}}, Invent. Math. 194 (2013), 73–118.

    • Crossref
    • Export Citation
  • [8]

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

  • [9]

    M. Coppens and G. Martens, Linear series on general k-gonal curves, Abh. Math. Semin. Univ. Hambg. 69 (1999), 347–371.

    • Crossref
    • Export Citation
  • [10]

    E. Coskun, Ulrich bundles on quartic surfaces with Picard number 1, C. R. Acad. Sci. Paris Sér. I 351 (2013), 221–224.

    • Crossref
    • Export Citation
  • [11]

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

  • [12]

    E. Coskun, R. Kulkarni and Y. Mustopa, The geometry of Ulrich bundles on del Pezzo surfaces, J. Algebra 375 (2013), 280–301.

    • Crossref
    • Export Citation
  • [13]

    L. Ein and R. Lazarsfeld, Stability and restrictions of Picard bundles with an application to the normal bundles of elliptic curves, Complex projective geometry, London Math. Soc. Lecture Note Ser. 179, Cambridge University Press, Cambridge (1992), 149–156.

  • [14]

    D. Eisenbud, G. Fløystad and F.-O. Schreyer, Sheaf cohomology and sheaf resolutions over exterior algebras, Trans. Amer. Math. Soc. 353 (2003), 4397–4426.

  • [15]

    D. Eisenbud, S. Popescu, F.-O. Schreyer and C. Walter, Exterior algebra methods for the minimal resolution conjecture, Duke Math. J. 112 (2002), 379–395.

    • Crossref
    • Export Citation
  • [16]

    D. Eisenbud and F.-O. Schreyer, Resultants and Chow forms via exterior syzygies, J. Amer. Math. Soc. 16 (2003), 537–579.

    • Crossref
    • Export Citation
  • [17]

    D. Eisenbud and F.-O. Schreyer, Boij–Söderberg theory, Combinatorial aspects of commutative algebra, Springer, Berlin (2011), 35–48.

  • [18]

    G. Farkas, M. Mustaţă and M. Popa, Divisors on g , g + 1 {\mathcal{M}_{g,g+1}} and the minimal resolution conjecture for points on canonical curves, Ann. Sci. Éc. Norm. Supér. (4) 36 (2003), 553–581..

    • Crossref
    • Export Citation
  • [19]

    M. Green, Koszul cohomology and the cohomology of projective varieties, J. Differential Geom. 19 (1984), 125–171.

    • Crossref
    • Export Citation
  • [20]

    M. Green and R. Lazarsfeld, Special divisors on curves on a K 3 {K3} surface, Invent. Math. 89 (1987), 357–370.

    • Crossref
    • Export Citation
  • [21]

    A. Hirschowitz and C. Simpson, La résolution minimale de l’arrangement d’un grand nombre de points dans 𝐏 n {\mathbf{P}^{n}}, Invent. Math. 126 (1996), 467–503.

  • [22]

    R. Lazarsfeld, A sampling of vector bundle techniques in the study of linear series, Lectures on Riemann surfaces (Trieste 1987), World Scientific, Teaneck (1989), 500–559.

  • [23]

    R. Lazarsfeld, Lectures on linear series, Complex algebraic geometry (Park City 1993), IAS/Park City Math. Ser. 3, American Mathematical Society, Providence (1997), 163–224.

  • [24]

    R. Miró-Roig and J. Pons-Llopis, The minimal resolution conjecture for points on del Pezzo surfaces, Algebra Number Theory 6 (2012), 27–46.

    • Crossref
    • Export Citation
  • [25]

    S. Mukai, Biregular classification of Fano 3-folds and Fano manifolds of coindex 3, Proc. Natl. Acad. Sci. USA 86 (1989), 3000–3002.

    • Crossref
    • Export Citation
  • [26]

    M. Mustaţă, Graded Betti numbers of general finite subsets of points on projective varieties, Matematiche (Catania) 53 (1998), 53–81.

  • [27]

    M. Popa, On the base locus of the generalized theta divisor, C. R. Acad. Sci. Paris Sér. I 329 (1999), 507–512.

    • Crossref
    • Export Citation
  • [28]

    M. Raynaud, Sections des fibrés vectoriels sur une courbe, Bull. Soc. Math. France 110 (1982), 103–125.

  • [29]

    E. Sernesi, On the existence of certain families of curves, Invent. Math. 75 (1984), 25–57.

    • Crossref
    • Export Citation
  • [30]

    C. Voisin, Green’s canonical syzygy conjecture for generic curves of odd genus, Compos. Math. 141 (2005), 1163–1190.

    • Crossref
    • Export Citation
Purchase article
Get instant unlimited access to the article.
$42.00
Log in
Already have access? Please log in.


or
Log in with your institution

Journal + Issues

The Journal für die reine und angewandte Mathematik is the oldest mathematics periodical still in existence. Founded in 1826 by August Leopold Crelle and edited by him until his death in 1855, it soon became widely known under the name of Crelle’s Journal. In the 190 years of its existence, Crelle’s Journal has developed to an outstanding scholarly periodical with one of the worldwide largest circulations among mathematics journals.

Search