Abstract
In its classical form, web geometry studies local configurations of finitely many smooth foliations in general position. In Sect. 1.1 the basic definitions of our subject are laid down and the algebraic webs are introduced. These are among the most important examples of the whole theory.
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Notes
- 1.
In other terms, the four base points of the pencils \(\mathcal{L}_{1},\ldots,\mathcal{L}_{4}\) are in ‘general position’ in \(\mathbb{P}^{2}\).
- 2.
Here, and throughout, i v denotes the interior product.
- 3.
The convention adopted in this text is that over a point p the fiber π −1(p) is the space of one-dimensional subspaces of \(T_{p}^{{\ast}}(\mathbb{C}^{n}, 0)\). Beware that some authors consider π −1(p) as the space of one-dimensional quotients of \(T_{p}^{{\ast}}(\mathbb{C}^{n}, 0)\).
- 4.
In case of confusion, notice that the coordinate functions on a vector space E can be chosen to be elements of E ∗, that is, linear forms on E.
- 5.
As usual the fundamental group acts on the right and thus \(\rho _{\mathcal{W}}\) is not a homomorphism but instead an anti-homomorphism, that is one has \(\rho _{\mathcal{W}}(\gamma _{1} \cdot \gamma _{2}) =\rho _{\mathcal{W}}(\gamma _{2}) \cdot \rho _{\mathcal{W}}(\gamma _{1})\) for arbitrary \(\gamma _{1},\gamma _{2} \in \pi _{1}(U,x_{0})\).
- 6.
When X is a singular variety, a web on X is a web on its smooth locus which extends to a global web on any of its desingularizations.
- 7.
Recall that a linear system is the projectivization \(\vert V \vert = \mathbb{P}(V )\) of a finite dimensional vector subspace \(V \subset H^{0}(S,\mathcal{L})\), where \(\mathcal{L}\) is a line-bundle on S. When S is a germ of surface, any line-bundle on it is trivial hence a linear system is nothing else than the projectivization of a finite dimensional vector space of germs of functions.
- 8.
An ordinary tacnode is a singularity of curve with exactly two branches, both of them smooth, having an ordinary tangency. Here tacnode refer to a curve cut out by a power series of the form
$$\displaystyle{\ell(x,y)^{2} +\ell (x,y)P_{ 2}(x,y) + h.o.t.}$$where ℓ is a linear form and P 2 is a homogeneous form of degree 2.
- 9.
This surface is a Del Pezzo surface of degree 5.
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Pereira, J.V., Pirio, L. (2015). Local and Global Webs. In: An Invitation to Web Geometry. IMPA Monographs, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-319-14562-4_1
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