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Nonabelian Jacobian J(X; L, d): Main Properties

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2072)

Abstract

In this section we introduce the main objects of our study and recall the main results of [R1].

Keywords

  • Hilbert Scheme
  • Smooth Part
  • Closed Subscheme
  • Free Sheaf
  • Invertible Sheaf

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Notes

  1. 1.

    As a section of \(\omega _{Z} \otimes \mathcal{O}_{X}(-L - K_{X})\).

  2. 2.

    The inclusion \({\pi }^{{\ast}}\big{(}pr_{2{\ast}}\big{(}\mathcal{J}_{\mathcal{Z}}\otimes pr_{1}^{{\ast}}\mathcal{O}_{X}(L + K_{X})\big{)}\big{)} \subset \mathbf{\tilde{F}}_{i}\) is proved in [R1], Proposition 1.6.

  3. 3.

    The self-duality of \(\tilde{\mathcal{F}}\) over Conf d (X) is provided by the quadratic form \(\mathbf{\tilde{q}}\) in (2.42).

References

  1. E. Arbarello, M. Cornalba, P. Griffiths, J. Harris, in Geometry of Algebraic Curves, vol. I. Grundlehren der Mathematischen Wissenschafen, 267 (Springer, New York, 1985)

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  2. I. Reider, Nonabelian Jacobian of smooth projective surfaces. J. Differ. Geom. 74, 425–505 (2006)

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  3. A.N. Tyurin, Cycles, curves and vector bundles on an algebraic surface. Duke Math. J. 54, 1–26 (1987)

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Reider, I. (2013). Nonabelian Jacobian J(X; L, d): Main Properties. In: Nonabelian Jacobian of Projective Surfaces. Lecture Notes in Mathematics, vol 2072. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35662-9_2

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