Connectedness of intersections of special Schubert varieties
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 dimension \(r\) on \(C\) has dimension equal to the Brill- Noether number \(((r + 1)d - rg - r(r + 1) + r)\) and is empty if the number is negative. The general curve of a given genus is \((d,r)\)-Brill- Noether for all \((d,r)\). Theorem. If the Brill-Noether number is strictly positive, then the varieties of special divisors of degree \(d\) and dimension \(r\) on a smooth, connected, \((d,r)\)-Brill-Noether curve \(C\) of genus \(g\) are connected.
connectedness of intersections of special Schubert varieties
scientific article; zbMATH DE number 617427