Abstract
So far, not many examples of abelian relations for webs appeared in this book. Besides the abelian relations for hexagonal 3-webs, the polynomial abelian relations for parallel webs (see Example 2.2.1), and the abelian relations for the planar quasi-parallel webs discussed in Example 2.2.4, which are by the way also polynomial, no other example was studied.
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Notes
- 1.
Here \(\Omega _{\mathbb{C}(Y )}\) stands for the module of Kähler differentials over \(\mathbb{C}(Y )\): by definition, it is the \(\mathbb{C}(Y )\)-module spanned by elements of the form \(d\varphi\) for \(\varphi \in \mathbb{C}(Y )\) subjected to the following relations: dc = 0 for every \(c \in \mathbb{C}\); and \(d(\phi \varphi ) =\phi d\varphi +\varphi d\phi\) for every \(\phi,\varphi \in \mathbb{C}(Y )\), see [70, II.§8].
- 2.
There is a subtlety here: a priori, [ω] is not intrinsically defined. According to [10], it is only defined up to the addition of a \(\overline{\partial }\)-closed current of type (1, 0) supported on the (finite) singular set of X (cf. [49] for a non-trivial example). In any case, the condition for [ω] to be \(\overline{\partial }\)-closed is well defined and intrinsic.
- 3.
In the formula, \(\chi (X,\mathcal{L})\) stands for the Euler-characteristic of the line-bundle \(\mathcal{L}\) which, by definition, is \(h^{0}(X,\mathcal{L}) - h^{1}(X,\mathcal{L})\).
- 4.
- 5.
Here, non-degenerate means that the curve is not contained in any abelian subvariety. Although this differs from our assumption on C, one of the first steps in the proof of this result is to establish that the maps PI used in the proof of Lemma 3.4.4 contain open subsets of A n in their images. Consequently, if C is non-degenerate, then the \(\mathcal{X}\) -dual web \(\mathcal{W}_{C}^{\mathcal{X}}\) is generically smooth.
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Pereira, J.V., Pirio, L. (2015). Abel’s Addition Theorem. In: An Invitation to Web Geometry. IMPA Monographs, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-319-14562-4_3
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