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Lazarsfeld–Mukai Bundles and Applications

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Commutative Algebra

Abstract

We survey the development of the notion of Lazarsfeld-Mukai bundles together with various applications, from the classification of Mukai manifolds to Brill-Noether theory and syzygies of K3 sections. To see these techniques at work, we present a short proof of a result of M. Reid on the existence of elliptic pencils.

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Notes

  1. 1.

    In fact, we do have a Harder–Narasimhan filtration, but we cannot control all the factors.

  2. 2.

    This ingenious procedure is an efficient replacement of the base-point-free pencil trick; “it has killed the base-point-free pencil trick,” to quote Enrico Arbarello.

  3. 3.

    Some authors consider that Mukai manifolds have dimension four or more.

  4. 4.

    In genus 11, it is actually birational [25].

  5. 5.

    The gonality gon(C) of a curve C is the minimal degree of a morphism from C to the projective line.

  6. 6.

    It is conjectured that the only other examples should be some half-canonical curves of even genus and maximal gonality [8]; however, this conjecture seems to be very difficult.

  7. 7.

    The indices p and q are usually forgotten when defining Koszul cohomology.

  8. 8.

    The dimension of K 1, q indicates the number of generators of degree (q + 1) in the homogeneous ideal.

  9. 9.

    Duality for Koszul cohomology of curves follows from Serre’s duality. For higher-dimensional manifolds, some supplementary vanishing conditions are required [11, 13].

  10. 10.

    Voisin’s and Teixidor’s cases complete each other quite remarkably.

  11. 11.

    A curvilinear subscheme is defined locally, in the classical topology, by \(x_{1} = \cdots = x_{s-1} = x_{s}^{k} = 0\); equivalently, it is locally embedded in a smooth curve.

  12. 12.

    The connectedness of X c [n] follows from the observation that a curvilinear subscheme is a deformation of a reduced subscheme.

  13. 13.

    We see one advantage of working on X c [n]: subtraction makes sense only for curvilinear subschemes.

  14. 14.

    The gonality for a singular stable curve is defined in terms of admissible covers [14].

  15. 15.

    Higher-rank Brill–Noether theory is a major, rapidly growing research field, and it deserves a separate dedicated survey.

  16. 16.

    For any line bundle A, we have γ(A  ⊕ n) = Cliff(A).

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Acknowledgements

I am grateful to S. Druel, G. Farkas, and A. Ortega for useful discussions on this subject. This work was partly supported by the grant “Vector bundle techniques in the geometry of complex varieties,” PN-II-ID-PCE-2011-3-0288, contract no. 132/05.10.2011.

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  1. Institute of Mathematics “Simion Stoilow”, Romanian Academy, 1-764, RO, 014700, Bucharest, Romania

    Marian Aprodu

  2. Şcoala Normală Superioară Bucureşti, Calea Griviţei 21, RO, 010702, Bucharest, Romania

    Marian Aprodu

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Correspondence to Marian Aprodu .

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  1. Cornell University, 547 Malott Hall, Ithaca, 14853, New York, USA

    Irena Peeva

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Aprodu, M. (2013). Lazarsfeld–Mukai Bundles and Applications. In: Peeva, I. (eds) Commutative Algebra. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5292-8_1

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