Hyperplanes of embeddable Grassmannians arise from projective embeddings: A short proof
Let \(n\geq 1\), \(\mathbb{K}\) be a field and \(k\in\{0,\dots, n-1\}\). Let \(V\) be an \((n+1)\)-dimensional vector space over \(\mathbb{K}\) and let \(PG(n,\mathbb{K})= PG(V)\). We can define the point-line geometry \(A_{n,k+1}\), where the points are \(k\)- dimensional subspaces of \(PG(n,\mathbb{K})\), the lines are sets \(A(\pi_1,\pi_2)\) of \(k\)-dimensional subspaces of \(PG(n,\mathbb{K})\) which contain subspaces \(\pi_1\) and are contained in a given \((k+1)\)-dimensional subspace \(\pi_2\) and incidence is containment. Let \(\Lambda^{k+1}V\) be the \((k+ 1)\)th exterior power of \(V\). For every \(k\)-dimensional subspace \(\alpha= \langle\overline v_1,\overline v_2,\dots,\overline v_{k+1}\rangle\) of \(PG(n,\mathbb{K})\), let \(e(\alpha)\) denote the point \(\langle\overline v_1\wedge\overline v_2\wedge\cdots\wedge\overline v_{k+1}\rangle\) of \(PG(\wedge^{k+ 1}V)\). The map \(e\) defines a full projective embedding of \(A_{n,k+1}\) into \(PG(\wedge^{k+1}V)\) which is called the Grassmann-embedding of \(A_{n,k+1}\). We can say that the hyperplane \(e^{-1}(\pi\cap e({\mathcal P}))\), where \({\mathcal P}\) is the point set of \(A_{n,k+1}\) arises from the Grassmann-embedding of \(A_{n,k+1}\). The aim of the present article is to give an alternative and shorter proof of a result of Shult: All hyperplanes of \(A_{n,k+1}\) arise from Grassmann-embedding of \(A_{n,k+1}\).
scientific article; zbMATH DE number 5376913