Connectedness of intersections of special Schubert varieties
type: Item
instance of: scholarly article
title: Connectedness of intersections of special Schubert varieties
zbMATH Open document ID: 0817.14001
DOI: 10.1007/BF02567610
author: V. Pereyra
published in: Manuscripta Mathematica
publication date: 1994-08-10T00:00:00Z
full work available at URL: https://eudml.org/doc/155957
review text: Let \(V\) be a vector space of dimension \(n\) on an algebraically closed field \(K\). Let \(G = Gr(\ell,V) = Gr(\ell, n)\) be the Grassmann variety of subspaces of \(V\) of dimension \(\ell\). If \(V'\) is a linear subspace of \(V\), we denote, as usual, by \(\sigma (V') = \{\Lambda \in G | \Lambda \cap V' \neq 0\}\) the special Schubert variety associated to \(V'\). We prove the following: Theorem. The intersection \(\bigcap \sigma (V_ j)\) of the special Schubert varieties associated to linear subspaces \(V_ j\), \(j = 1, \dots, m\), of dimension \(n - \ell - a_ j + 1\) such that \(\ell (n - \ell) - \sum^{j=m}_{j=1} a_ j > 0\), is connected. Moreover, the intersection is irreducible of dimension \(\ell (n - \ell) - \sum a_ j\) for a general choice of the subspaces \(V_ j\). The second part of the theorem follows from its first part and the local irreducibility. \textit{J. Hansen} [Am. J. Math. 105, 633-639 (1983; Zbl 0544.14034)] proved a result about the connectedness of intersections of subvarieties of Grassmannians and flag varieties, but under the stronger hypothesis that the expected codimension of the intersection is smaller than \(n - 1\). As initial motivation for this work, let us remark that a particular case \((\dim V_ j = 2\) for all \(j)\) of this statement appears, without a proof, in a book by \textit{F. Enriques} and \textit{O. Chisini} [``Lezioni sulla teoria geometrica delle equazioni: e delle funzioni algebriche'' Vol. III (1924; reprint 1985; Zbl 0571.51001), page 524]. We conjecture that the irreducibility holds for intersections of Schubert varieties (not necessarily special), when they are in general position with nonempty intersection. (We have checked the conjecture for the case of two Schubert varieties.) As an example of application we provide a new proof (over the field of complex numbers) of the following theorem. Recall that a curve of genus \(g\) is said \((d,r)\)-Brill-Noether if the variety of linear series of degree \(d\) and dim…
Mathematics Subject Classification ID: 14C17
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 14F45
zbMATH DE Number: 617427
zbMATH Keywords: connectedness of intersections of special Schubert varieties
zbMATH Keywords: \((d,r)\)- Brill-Noether curve
MaRDI profile type: Publication
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OpenAlex ID: W1984466237
description: scientific article; zbMATH DE number 617427