Primitive multiple schemes

Abstract

A primitive multiple scheme is a Cohen–Macaulay scheme Y such that the associated reduced scheme \(X=Y_{\mathrm{red}}\) is smooth, irreducible, and Y can be locally embedded in a smooth variety of dimension \(\dim (X)+1\). If n is the multiplicity of Y, there is a canonical filtration \(X=X_1\subset X_2\subset \cdots \subset X_n=Y\), such that \(X_i\) is a primitive multiple scheme of multiplicity i. The simplest example is the trivial primitive multiple scheme of multiplicity n associated to a line bundle L on X: it is the nth infinitesimal neighborhood of X, embedded in the line bundle \(L^*\) by the zero section. Let . The primitive multiple schemes of multiplicity n are obtained by taking an open cover \((U_i)\) of a smooth variety X and by gluing the schemes \(U_i\,{\times }\, \mathbf{Z}_n\) using automorphisms of \(U_{ij}\,{\times }\, \mathbf{Z}_n\) that leave \(U_{ij}\) invariant. This leads to the study of the sheaf of nonabelian groups of automorphisms of \(X\,{\times }\, \mathbf{Z}_n\) that leave the X invariant, and to the study of its first cohomology set. If \(n\geqslant 2\) there is an obstruction to the extension of \(X_n\) to a primitive multiple scheme of multiplicity \(n+1\), which lies in the second cohomology group \(H^2(X,E)\) of a suitable vector bundle E on X. In this paper we study these obstructions and the parametrization of primitive multiple schemes. As an example we show that if \(X={{\mathbb {P}}}_m\) with \(m\geqslant 3\), all the primitive multiple schemes are trivial. If \(X={{\mathbb {P}}}_2\), there are only two nontrivial primitive multiple schemes, of multiplicities 2 and 4, which are not quasi-projective. On the other hand, if X is a projective bundle over a curve, we show that there are infinite sequences \( X=X_1\subset X_2\subset \cdots \subset X_n\subset X_{n+1}\subset \cdots \) of nontrivial primitive multiple schemes.

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References

  1. 1.

    Bangere, P., Gallego, F.J., González, M.: Deformations of hyperelliptic and generalized hyperelliptic polarized varieties (2020). arXiv:2005.00342

  2. 2.

    Bangere, P., Mukherjee, J., Raychaudhury, D.: K3 carpets on minimal rational surfaces and their smoothing (2020). arXiv:2006.16448

  3. 3.

    Bănică, C., Forster, O.: Multiplicity structures on space curves. In: Sundararaman, D. (ed.) The Lefschetz Centennial Conference, Part I. Contemporary Mathematics, vol. 58, pp. 47–64. American Mathematical Society, Providence (1986)

  4. 4.

    Bayer, D., Eisenbud, D.: Ribbons and their canonical embeddings. Trans. Amer. Math. Soc. 347(3), 719–756 (1995)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Chen, D., Kass, J.L.: Moduli of generalized line bundles on a ribbon. J. Pure Appl. Algebra 220(2), 822–844 (2016)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Drézet, J.-M.: Déformations des extensions larges de faisceaux. Pacific J. Math. 220(2), 201–297 (2005)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Drézet, J.-M.: Faisceaux cohérents sur les courbes multiples. Collect. Math. 57(2), 121–171 (2006)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Drézet, J.-M.: Paramétrisation des courbes multiples primitives. Adv. Geom. 7(4), 559–612 (2007)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Drézet, J.-M.: Faisceaux sans torsion et faisceaux quasi localement libres sur les courbes multiples primitives. Math. Nachr. 282(7), 919–952 (2009)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Drézet, J.-M.: Sur les conditions d’existence des faisceaux semi-stables sur les courbes multiples primitives. Pacific J. Math. 249(2), 291–319 (2011)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Drézet, J.-M.: Courbes multiples primitives et déformations de courbes lisses. Ann. Fac. Sci. Toulouse Math. 22(1), 133–154 (2013)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Drézet, J.-M.: Fragmented deformations of primitive multiple curves. Cent. Eur. J. Math. 11(12), 2106–2137 (2013)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Drézet, J.-M.: Reducible deformations and smoothing of primitive multiple curves. Manuscripta Math. 148(3–4), 447–469 (2015)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Drézet, J.-M.: Reachable sheaves on ribbons and deformations of moduli spaces of sheaves. Int. J. Math. 28(12), # 1750086 (2017)

  15. 15.

    Eisenbud, D., Green, M.: Clifford indices of ribbons. Trans. Amer. Math. Soc. 347(3), 757–765 (1995)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Ferrand, D.: Courbes gauches et fibrés de rang 2. C. R. Acad. Sci. Paris Sér. A-B 281(10), 345–347 (1975)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Fong, L.-Y.: Rational ribbons and deformation of hyperelliptic curves. J. Algebraic Geom. 2(2), 295–307 (1993)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Frenkel, J.: Cohomologie non abélienne et espaces fibrés. Bull. Soc. Math. France 85, 135–220 (1957)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Gallego, F.J., González, M., Purnaprajna, B.P.: Deformation of finite morphisms and smoothing of ropes. Compositio Math. 144(3), 673–688 (2008)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Gallego, F.J., González, M., Purnaprajna, B.P.: \(K3\) double structures on Enriques surfaces and their smoothings. J. Pure Appl. Algebra 212(5), 981–993 (2008)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Gallego, F.J., González, M., Purnaprajna, B.P.: An infinitesimal condition to smooth ropes. Rev. Mat. Complut. 26(1), 253–269 (2013)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Godement, R.: Topologie algébrique et théorie des faisceaux. Actualités scientifiques et industrielles, vol. 1252. Hermann, Paris (1964)

  23. 23.

    González, M.: Smoothing of ribbons over curves. J. Reine Angew. Math. 591, 201–235 (2006)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Grothendieck, A., et al.: Revêtements Étales et Groupe Fondamental. SGA1. Lecture Notes in Mathematics, vol. 224. Springer, Berlin (1971)

  25. 25.

    Hartshorne, R.: Ample vector bundles. Inst. Hautes Études Sci. Publ. Math. 29, 63–94 (1966)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, Berlin (1977)

  27. 27.

    Savarese, M.: Coherent sheaves on ribbons and their moduli (2019). arXiv:1902.08510

  28. 28.

    Savarese, M.: Generalized line bundles on primitive multiple curves and their moduli (2019). arXiv:1902.09463

  29. 29.

    Savarese, M.: On the irreducible components of the compactified Jacobian of a ribbon. Comm. Algebra 47(4), 1385–1389 (2019)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

I am grateful to the anonymous referee for giving some useful remarks.

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Correspondence to Jean-Marc Drézet.

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Drézet, JM. Primitive multiple schemes. European Journal of Mathematics (2021). https://doi.org/10.1007/s40879-020-00447-4

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Keywords

  • Multiple structures
  • Obstructions
  • Sheaves of nonabelian groups

Mathematics Subject Classification

  • 14D20
  • 14B20