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Notes
- 1.
The Grassmannian \(\mathrm{Grass}(V,r)\) is isomorphic to the projectivization of the image of the multilinear map
$$\displaystyle\begin{array}{rcl} \varphi: V ^{r}& --\rightarrow &\bigwedge ^{r}V \, {}\\ (v_{1},\ldots,v_{r})& \mapsto & v_{1} \wedge \cdots \wedge v_{r}\,. {}\\ \end{array}$$The isomorphism is given of course by associating with any W ∈ Grass(V, r) the point \([\varphi (w_{1},\ldots,w_{r})] \in \mathbb{P}(\wedge ^{r}V )\) where w 1, …, w r is an arbitrary basis of W. Clearly, \([\varphi (w_{1},\ldots,w_{r})]\) does not depend on the chosen basis. The induced map \(\mathrm{Grass}(V,r) \rightarrow \mathbb{P}\left (\wedge ^{r}V \right )\) is the so-called Plücker embedding of \(\mathrm{Grass}(V,r)\).
- 2.
Here the abelian relations η (j) are seen as coordinate functions on \(\mathcal{A}_{2}(\mathcal{W})\), which is the same as thinking of them as elements of \(\mathcal{A}_{2}(\mathcal{W})^{{\ast}}\).
- 3.
This is just the set of morphisms from \(\mathbb{P}^{1}\) to \(\mathbb{P}^{n}\) of degree k − n − 1 which can be naturally identified with a Zariski open subset of \(\mathbb{P}\left (\mathbb{C}_{k-n-1}[s,t]^{\pi }\right )\).
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Pereira, J.V., Pirio, L. (2015). Algebraization of Maximal Rank Webs. In: An Invitation to Web Geometry. IMPA Monographs, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-319-14562-4_5
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