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 \(\mathcal{T}_{\pi }\)).
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Notes
- 1.
See e.g., Mumford’s survey, [Mu], and the references therein for more details.
- 2.
This holds if \(\mathcal{E}_{[\alpha ]}\) is, for example, generated by its global sections.
- 3.
“Vertical” as usual refers to being tangent along the fibres of the natural projection \(\pi : \mathbf{\breve{J}}_{\Gamma }\rightarrow \breve{\Gamma }\).
- 4.
If no ambiguity is likely, the parameter t will be omitted from the above notation.
- 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.
The number of boxes erased is r + 1, the dimension of H 0([Z], [α]).
- 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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