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:

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 without using a presentation of $\mathcal{E}$.

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 $\mathcal{O}_{\mathbb{P}}(1)$ be the universal invertible sheaf on $\mathbb{P}(\wedge^d \mathcal{E})$ together with its universal epimorphism $s : \wedge^d \mathcal{E} \otimes_{\mathcal{O}_S} \mathcal{O}_{\mathbb{P}} \twoheadrightarrow \mathcal{O}_{\mathbb{P}}(1)$ . Then define

$P : \wedge^{d-1}(\mathcal{E}) \otimes_{\mathcal{O}_S} \wedge^{d+1}(\mathcal{E}) \otimes_{\mathcal{O}_S} \mathcal{O}_{\mathbb{P}} \to \mathcal{O}_{\mathbb{P}}(2),$ $${\small f_1 \wedge \dotsc \wedge f_{d-1} \otimes e_0 \wedge \dotsc \wedge e_d \mapsto \sum_{l=0}^{d} (-1)^l s(f_1 \wedge \dotsc \wedge f_{d-1} \wedge e_k) \otimes s(e_0 \wedge \dotsc \wedge \widehat{e_k} \wedge \dotsc \wedge e_d).}$$

Then $I$ is the image of $\check{P} : \wedge^{d-1}(\mathcal{E}) \otimes_{\mathcal{O}_S} \wedge^{d+1}(\mathcal{E}) \otimes_{\mathcal{O}_S} \mathcal{O}_{\mathbb{P}}(-2) \to \mathcal{O}_{\mathbb{P}}$.

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 $S$.

  • 3
    $\begingroup$ Note that I am not looking for the equations in the classical special case $\mathrm{Grass}_d(R^n) \hookrightarrow \mathbb{P}(\wedge^d R^n)$. $\endgroup$ – Martin Brandenburg Dec 13 '12 at 9:48

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.

V. Lakshmibai & K.N. Raghavan have written a book published by Springer (2008). Encyclopaedia of Mathematical Sciences, 137.

  • $\begingroup$ You might start with Seshadri's "Standard Monomial Theory -- A Historical Account" (in volume 2 of his collected works) or Lakshmibai/Littelmann/Magyar's "Standard Monomial Theory and Applications" (available online). $\endgroup$ – Michael Joyce Dec 13 '12 at 12:39
  • $\begingroup$ They only deal with the classical case, i.e. $S=\mathrm{Spec}(k)$ for a field $k$ and $\mathcal{E}=k^n$. $\endgroup$ – Martin Brandenburg Dec 13 '12 at 17:31

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, Classification of Higher Dimensional Algebraic Varieties.

  • $\begingroup$ Thank you, but this only contains the special case that $\mathcal{E}$ is locally free of finite rank. $\endgroup$ – Martin Brandenburg Dec 13 '12 at 17:32

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