By Israel M. Gelfand, Mikhail Kapranov, Andrei Zelevinsky
"This ebook revives and drastically expands the classical concept of resultants and discriminants. many of the major new result of the ebook were released prior in additional than a dozen joint papers of the authors. The publication properly enhances those unique papers with many examples illustrating either previous and new result of the theory."—Mathematical Reviews
"Collecting and increasing the basic and hugely unique result of the authors, it provides a special mixture of classical arithmetic and intensely contemporary advancements in algebraic geometry, homological algebra, and combinatorial theory." —Zentralblatt Math
"This ebook is very instructed a good way to get into the thick of up to date algebra, or if you want to locate a few attention-grabbing challenge to paintings on, whose resolution will profit mankind." —Gian-Carlo Rota, Advanced publication Reviews
"…the publication is sort of completely written, and therefore I warmly suggest it not just to students yet specifically to scholars. The latter do want a textual content with broader perspectives, which indicates that arithmetic is not only a series of it sounds as if unrelated expositions of recent theories, … yet in its place a truly large and complex construction whose edification may possibly occasionally adventure problems … yet ultimately progresses steadily." —Bulletin of the yankee Mathematical Society
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Extra info for Discriminants, Resultants, and Multidimensional Determinants
Example text
So both of these varieties are smooth and have codimension n in their ambient projective spaces. In particular, this provides an example of a smooth subvafiety whose dual is not a hypersurface. 14. Let X be the product of three projective spaces in its Segre em- bedding: X -- P ( E ) x P ( F ) x P ( G ) ~ P ( E | F | G), (e, f, g) ~ e | f | g. By choosing bases in E, F, G, we can identify the space E | F | G as well as its dual, with the space of 3-dimensional matrices (arrays) A = Ilaijk II where i -- 1.
6. A projective variety X C pn is called non-degenerate if it does not lie in a hyperplane. A variety X is called linearly normal if it is non-degenerate and cannot be represented as an isomorphic projection of a non-degenerate variety from a projective space of higher dimension. The embedding of a linearly normal variety can be described intrinsically, in terms of the variety itself and a certain invertible sheaf on it. Let us recall this correspondence between invertible sheaves and projective embeddings [GH] [Hart].
The converse statement is crucial for us. 1. Any conic Lagrangian subvariety A C T*M has the form Con(Z) for some irreducible subvariety Z C M. Proof Let pr 9 T*M --+ M be the projection. Define Z = pr(A). This is an irreducible subvariety in M. Let z be any smooth point of Z. We claim that the fiber pr -1 (z) N A is contained in the conormal space (T~M)z. Indeed, let ~ be any covector in pr -1 (z) fq A. Since Tz M is a vector space, we can regard ~ as a tangent vector to T* M at a point z 6 M C T* M (here we identify M with the zero section of the projection pr : T*M --+ M).