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Grassmann bundles (1981)

Scott, D. B.

Springer

Annali di matematica pura ed applicata 127 (1981), S. 101-140

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ISSN: 1618-1891

Source: Springer Online Journal Archives 1860-2000

Topics: Mathematics

Notes: Summary In the classical theory of the Grassmann Variety there are three principal results. The Basis Theorem asserts that the Chow ring has a selfdual linear basis of $$\left( {\mathop d\limits^n } \right)$$ classes. Determinantal Formulawhich expresses any basic class as a determinant in the special classes. Finally the ring structure is elucidated by Pieri's Formulawhich expresses the intersection of a basic class and a special class in terms of the basic classes. Here we show how all these results can be established also for the Chow ring of a Grassmann bundle. There are however some differences. In the classical case the basic classes are Schubert classes: this is impossible in the general case as there need not be enough Schubert classes to provide a basis and in the general case there is a pair of dual bases which both reduce to the Schubert basis in the classical case. In addition to these generalizations of the classical results we also enlarge on the theory of Schubert classes developed in the important paper of Kempf and Laksov [4].Following them we shall henceforth use the phrase « determinantal formula » to mean their formula for Schubert classes and our generalization of it to « improper » Schubert classes.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1007/BF01811721

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