Complete collineations and blowing up determinantal ideals
Let \(E=E_ 0\supset E_ 1\supset...\supset E_ t\) and \(F=F_ 0\twoheadrightarrow F_ 1\twoheadrightarrow...\twoheadrightarrow F_ t\) be flags of vector spaces. Call a complete collineation from E to F to be a sequence of linear maps \(v_ i:E_ i\to F_ i (i=0,...,t)\) up to scalar multiple such that \(E_ i=\ker v_{i-1}\), \(F_ i=co\ker v_{i- 1}\) and \(v_ t\) is injective \(\neq 0\). The sequence \((n+1\)-rank \(E_ i)\) where \(n+1=rank E\), is called the type of \((v_ i)\). Let \(S_ r\) denote the projective space of linear maps \(\bigwedge^{r+1}E\to \bigwedge^{r+1}F.\) Let \(S^ t\) be the closure of the set of all \((v,\bigwedge^{2}v,...,\bigwedge^{t+1}v)\) in \(S_ 0\times...\times S_ t\). It is shown that \(S^ t\) is smooth and equal to the successive blow-up of \(S_ 0\) along the loci of maps of rank \(\leq t\). The set of complete collineations of a given type is shown to be naturally parametrized by an orbit of \(S^ n\) under GL(E)\(\times GL(F)\) and the closure of any such is a transversal intersection of divisors. The Chow ring of \(S^ n\) is described and some applications to the enumerative geometry of collineations are given. It is shown that the coefficients appearing in a formula of A. Lascoux for the Segre classes of a tensor product of vector bundles are the solutions of a simple enumerative problem for collineations as stated by \textit{D. Laksov} in Enumerative Geometry and Classical Algebraic Geometry, Prog. Math. 24, 107-132 (1982; Zbl 0501.14031); this volume contains also the author's article ''Schubert calculus for complete quadrics'' [Prog. Math. 24, 199-235 (1982; Zbl 0501.14032)], where similar results are obtained for symmetric linear maps. Related material may be found in the following papers: \textit{C. De Concini} and \textit{C. Procesi}, ''Complete symmetric varieties'' (Prepr. Univ. Roma, 1982) and \textit{D. Luna} and \textit{T. Vust}, ''Plongements d'espaces homogènes'', Comm. Math. Helv. 58, 186-245 (1983).
scientific article; zbMATH DE number 3865488
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