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Elliptic curves, ACM bundles and Ulrich bundles on prime Fano threefolds

  • Ciliberto, Ciro [2] ; Flamini, Flaminio [2] ; Knutsen, Andreas Leopold [1]
    1. [1] University of Bergen

      University of Bergen

      Noruega

    2. [2] Ciro Ciliberto, Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 00173, Roma, Italy
  • Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 75, Fasc. 3, 2024, págs. 795-822
  • Idioma: inglés
  • DOI: 10.1007/s13348-023-00413-9
  • Enlaces
  • Resumen

    • Let X be any smooth prime Fano threefold of degree 2g-2 in {\mathbb P}^{g+1}, with g \in \{3,\ldots ,10,12\}. We prove that for any integer d satisfying \left\lfloor \frac{g+3}{2} \right\rfloor \leqslant d \leqslant g+3 the Hilbert scheme parametrizing smooth irreducible elliptic curves of degree d in X is nonempty and has a component of dimension d, which is furthermore reduced except for the case when (g,d)=(4,3) and X is contained in a singular quadric. Consequently, we deduce that the moduli space of rank–two slope–stable ACM bundles {\mathcal F}_d on X such that \det ({\mathcal F}_d)={\mathcal O}_X(1) , c_2({\mathcal F}_d)\cdot {\mathcal O}_X(1)=d and h^0({\mathcal F}_d(-1))=0 is nonempty and has a component of dimension 2d-g-2 , which is furthermore reduced except for the case when (g,d)=(4,3) and X is contained in a singular quadric. This completes the classification of rank–two ACM bundles on prime Fano threefolds. Secondly, we prove that for every h \in {\mathbb Z}^+ the moduli space of stable Ulrich bundles {\mathcal E} of rank 2h and determinant {\mathcal O}_X(3h) on X is nonempty and has a reduced component of dimension h^2(g+3)+1 ; this result is optimal in the sense that there are no other Ulrich bundles occurring on X. This in particular shows that any prime Fano threefold is Ulrich wild.

  • Referencias bibliográficas
    • Arrondo, E., Costa, L.: Vector bundles on Fano 3-folds without intermediate cohomology. Comm. Algorithm 28, 3899–3911 (2000)
    • Arrondo, E.: A home-made Hartshorne-Serre correspondence. Rev. Mat. Complut. 20, 423–443 (2007)
    • Beauville, A.: An introduction to Ulrich bundles. Eur. J. Math. 4, 26–36 (2018)
    • Beauville, A.: Determinantal hypersurfaces. Michigan Math. J. 48, 39–64 (2000)
    • Beauville, A.: Vector bundles on Fano threefolds and K3 surfaces. Boll. Unione Mat. Ital. 15, 43–55 (2022)
    • Beauville, A.: Vector bundles on the cubic threefold, Contemp. Math. 312, Amer. Math. Soc., Providence, RI, 71–86 (2002)
    • Brambilla, M.C., Faenzi, D.: Moduli spaces of rank-2 ACM bundles on prime Fano threefolds. Michigan Math. J. 60, 113–148 (2011)
    • Brambilla, M.C., Faenzi, D.: Rank 2 stable sheaves with odd determinant on Fano threefolds of genus 9. Math. Z. 275, 185–210 (2013)
    • Brambilla, M.C., Faenzi, D.: Vector bundles on Fano threefolds of genus 7 and Brill-Noether loci. Internat. J. Math. 25, 1450023 (2014)
    • Casnati, G., Faenzi, D., Malaspina, F.: Moduli spaces of rank two aCM bundles on the Segre product of three projective lines. J. Pure Appl...
    • Casnati, G., Faenzi, D., Malaspina, F.: Rank two aCM bundles on the del Pezzo fourfold of degree 6 and its general hyperplane section. J....
    • Casnati, G., Faenzi, D., Malaspina, F.: Rank 2 ACM bundles on the del Pezzo threefold of Picard number 3. J. Algebra 429, 413–446 (2015)
    • Casnati, G., Filip, M., Malaspina, F.: Rank two aCM bundles on the del Pezzo threefold of degree 7. Rev. Math Complutense 37, 129–165 (2017)
    • Casanellas, M., Hartshorne, R.: with an appendix by F. Geiss and F.-O. Schreyer, Stable Ulrich bundles. Internat. J. Math. 23, 1250083, 50...
    • Cho, Y., Kim Y., Lee, K.-S.: Ulrich bundles on intersection of two 4-dimensional quadrics. Int. Math. Res. Notices 2021(22), 17277–17303 (2021)
    • Ciliberto, C., Flamini, F., Knutsen, A.L.: Ulrich bundles on Del Pezzo threefolds. J. Algebra 634, 209–236 (2023)
    • Coskun, E.: A Survey of Ulrich bundles. In: Analytic and Algebraic Geometry, Springer, pp. 86–106 (2017)
    • Costa, L., Miró-Roig, M.R.: Brill-Noether theory for moduli spaces of sheaves on algebraic varieties. Forum Math. 22, 411–432 (2010)
    • Costa, L., Miró–Roig, R. M., Pons–Llopis, J.: Ulrich bundles from commutative algebra to algebraic geometry, De Gruyter. Stud. Math. 77 (2021)
    • Drozd, Y., Greuel, G.M.: Tame and wild projective curves and classification of vector bundles. J. Algebra 246, 1–54 (2001)
    • Fanography, https://www.fanography.info/
    • Feyzbakhsh, S., Pertusi, L.: Serre-invariant stability conditions and Ulrich bundles on cubic threefolds, Épijournal de Géométrie Algébrique...
    • Harris, J.: Theta-characteristics on algebraic curves. Trans. Am. math. Soc. 271, 611–638 (1982)
    • Hartshorne, R.: Stable reflexive sheaves. Math. Ann. 254, 121–176 (1980)
    • Hartshorne, R.: Stable vector bundles of rank 2 on ^{3}. Math. Ann. 238, 229–280 (1978)
    • Hartshorne, R., Hirschowitz, A.: Smoothing algebraic space curves. Algebraic Geometry, Sitges, pp. 98–131; Lecture Notes in Math. 1124, Springer,...
    • Harris, J., Tschinkel, Y.: Rational points on quartics. Duke Math. J. 104, 477–500 (2000)
    • Iskovskih, V.A.: Fano 3-folds. II, Izv. Akad. Nauk SSSR Ser. Mat. 42, 506–549 (1978)
    • Iskovskih, V.A.: Double projection from a line on Fano threefolds of the first kind. Math. USSR Sb. 66, 265–284 (1990)
    • Iskovskih, V.A., Prokhorov, Yu. G.: it Algebraic Geometry V: Fano varieties. Encyclopaedia of Math. Sciences 47, Springer-Verlag, Berlin (1999)
    • Knutsen, A.L.: Smooth, isolated curves in families of Calabi-Yau threefolds in homogeneous spaces. J. Korean Math. Soc. 50, 1033–1050 (2013)
    • Lazarsfeld, R.: Brill-Noether-Petri without degenerations. J. Differential Geom. 23, 299–307 (1986)
    • Lahoz, M., Macrì, E., Stellari, P.: Arithmetically Cohen-Macaulay bundles on cubic threefolds. Algebraic. Geometry 2, 231–269 (2015)
    • Lee, K.-S., Park, K.-D.: Moduli spaces of Ulrich bundles on the Fano 3-fold V_5. J. Algebra 574, 262–277 (2021)
    • Markushevich, D., Tikhomirov, A.S.: The Abel-Jacobi map of a moduli component of vector bundles on the cubic threefold. J. Algebraic Geom....
    • Mukai, S.: New development of theory of Fano 3-folds: vector bundle method and moduli problem, Sugaku 47 (1995), 125–144; translation. Sugaku...
    • Mukai, S., Umemura, H.: Minimal rational threefolds, Lecture Notes in Math. 1016, Springer-Verlag , pp. 490–518 (1983)
    • Russo, F.: On the geometry of some special projective varieties, UMI–Springer Lecture Notes in Math. 18, (2016)
    • Voisin, C.: Hodge Theory and Complex Algebraic Geometry, II, Cambridge Studies in Adv. Math. 77, Cambridge, (2007)
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