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Configurations and Theirs Equations

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

Abstract

In previous sections we have seen how the nilpotent elements of \(\boldsymbol{\mathcal{G}}_{\mbox{ $\Gamma $}}\), given by the values of morphisms d ± [see (5.26) and (5.35)], give rise to very rich algebraic and geometric structures on J(X; L, d) (to be more precise on the relative tangent sheaf \(\mathcal{T}_{\pi }\)).

Keywords

  • Line Bundle
  • Projective Space
  • Complete Intersection
  • Global Section
  • Nilpotent Element

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 1.

    See e.g., Mumford’s survey, [Mu], and the references therein for more details.

  2. 2.

    This holds if \(\mathcal{E}_{[\alpha ]}\) is, for example, generated by its global sections.

  3. 3.

    “Vertical” as usual refers to being tangent along the fibres of the natural projection \(\pi : \mathbf{\breve{J}}_{\Gamma }\rightarrow \breve{\Gamma }\).

  4. 4.

    If no ambiguity is likely, the parameter t will be omitted from the above notation.

  5. 5.

    Recall: \({\mathbf{H}}^{0}([Z], [\alpha ]) = \mathbf{\tilde{H}}([Z], [\alpha ])\) and \(dim\mathbf{\tilde{H}}([Z], [\alpha ]) = r + 1\) is the index of L-speciality of Z. Hence the upper bound of the interval of [1, r + 1]. The lower bound comes from the fact that D  + (t) annihilates the subspace \(\mathbf{C}\{1_{Z}\} \subset \mathbf{\tilde{H}}([Z], [\alpha ])\) of constant functions.

  6. 6.

    The number of boxes erased is r + 1, the dimension of H 0([Z], [α]).

  7. 7.

    The d  − -multiplicity \(\mu _{(l_{\Gamma }-1)0}^{{\prime}}(v)\) counts precisely the number of rows of λ(v) consisting of a single box having degree 0.

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Reider, I. (2013). Configurations and Theirs Equations. 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_10

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