What is the ideal corresponding to the Plücker embedding? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T01:48:22Z http://mathoverflow.net/feeds/question/116249 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116249/what-is-the-ideal-corresponding-to-the-plucker-embedding What is the ideal corresponding to the Plücker embedding? Martin Brandenburg 2012-12-13T07:55:32Z 2012-12-13T10:03:49Z <p>Let $S$ be a noetherian scheme, $\mathcal{E}$ a quasi-coherent sheaf on $S$ and let $d \in \mathbb{N}$. There is a Plücker embedding $\omega : \mathrm{Grass}_d(\mathcal{E}) \hookrightarrow \mathbb{P}(\wedge^d \mathcal{E})$. A very elegant functorial construction can be found in EGA I, 9.8. My question is:</p> <p>How can we describe the corresponding quasi-coherent ideal $I$ on $\mathbb{P}(\wedge^d \mathcal{E})$ globally? More precisely, if $\mathcal{E}$ is coherent, then by results of EGA II there is an epimorphism $\oplus_i M_i \otimes_{\mathcal{O}_S} \mathcal{O}(n_i) \twoheadrightarrow I$ for some coherent $\mathcal{O}_S$-modules $M_i$ and integers $n_i$. I would like know if one can write this down <em>without</em> using a presentation of $\mathcal{E}$.</p> <p>The answer in the special case $\mathcal{E} = \mathcal{O}_S^{\oplus I}$ for some set $I$ is well-known (at least when $S$ is a field and $I$ is finite, but the general case works the same. Does anybody know a reference where this is done?): The Plücker relations generate $I$. More precisely, let <code>$\mathcal{O}_{\mathbb{P}}(1)$</code> be the universal invertible sheaf on $\mathbb{P}(\wedge^d \mathcal{E})$ together with its universal epimorphism <code>$s : \wedge^d \mathcal{E} \otimes_{\mathcal{O}_S} \mathcal{O}_{\mathbb{P}} \twoheadrightarrow \mathcal{O}_{\mathbb{P}}(1)$</code> . Then define</p> <p><code>$P : \wedge^{d-1}(\mathcal{E}) \otimes_{\mathcal{O}_S} \wedge^{d+1}(\mathcal{E}) \otimes_{\mathcal{O}_S} \mathcal{O}_{\mathbb{P}} \to \mathcal{O}_{\mathbb{P}}(2),$</code> $${\small f_1 \wedge \dotsc \wedge f_{d-1} \otimes e_0 \wedge \dotsc \wedge e_d \mapsto \sum_{l=0}^{d} (-1)^l s(f_1 \wedge \dotsc \wedge f_{d-1} \wedge e_k) \otimes s(e_0 \wedge \dotsc \wedge \widehat{e_k} \wedge \dotsc \wedge e_d).}$$</p> <p>Then $I$ is the image of <code>$\check{P} : \wedge^{d-1}(\mathcal{E}) \otimes_{\mathcal{O}_S} \wedge^{d+1}(\mathcal{E}) \otimes_{\mathcal{O}_S} \mathcal{O}_{\mathbb{P}}(-2) \to \mathcal{O}_{\mathbb{P}}$</code>.</p> <p>For general $\mathcal{E}$, these Plücker relations are also satisfied, but I couldn't prove the converse and meanwhile I'm convinced that we need more relations. If it helps, you may assume that $2$ is invertible on $S$.</p> http://mathoverflow.net/questions/116249/what-is-the-ideal-corresponding-to-the-plucker-embedding/116252#116252 Answer by Sándor Kovács for What is the ideal corresponding to the Plücker embedding? Sándor Kovács 2012-12-13T08:29:58Z 2012-12-13T08:29:58Z <p>I am not sure if it fits what you are looking form but there is a sort of a description of this ideal sheaf in section 12.A in Hacon-Kovács, <a href="http://books.google.com/books?id=ZUy_uMu2P4MC&amp;source=gbs_navlinks_s" rel="nofollow">Classification of Higher Dimensional Algebraic Varieties</a>. </p> http://mathoverflow.net/questions/116249/what-is-the-ideal-corresponding-to-the-plucker-embedding/116253#116253 Answer by P Vanchinathan for What is the ideal corresponding to the Plücker embedding? P Vanchinathan 2012-12-13T08:36:09Z 2012-12-13T08:36:09Z <p>References for this purpose are: A series of papers initiated by C S Seshadri, Lakshmibai, Musili develops "Standard Monomial Theory" to deal with this. It gives equations for Schubert varietes, describes their singular loci, proves many cohomology-vanishing theorems for line bundles on them.</p> <p>V. Lakshmibai &amp; K.N. Raghavan have written a book published by Springer (2008). Encyclopaedia of Mathematical Sciences, 137. </p>