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Lectures on results on Bezout’s theorem. Notes by D. P. Patil. (English) Zbl 0553.14022

The lectures are mainly on the case of two pure dimensional homogeneous ideals in a polynomial ring and study the product of their degrees expressed as a sum of local contributions: \(\deg (X) \deg (Y) \sum_{i}j(X,Y;C_ i).\deg (C_ i)\). Here \(X\) and \(Y\) are the two given (not necessarily reduced) varieties in the projective space and the \(C_ i\) are certain subvarieties of \(X\cap Y\) of dimensions at least the proper one. (The occurring \(C_ i\) are all the irreducible components of the intersection and furthermore “embedded” components, but not in the usual sense; one empty \(C_ i\) may also occur.) Their multiplicities \(j(X,Y;C_ i)\) are given as the lengths of corresponding primary ideals in a polynomial ring over a higher dimensional ground field extension, which are inductively defined. This main theorem of Stückrad and Vogel is proven in detail.
There is an extensive historical survey as well as a list of 106 references and a reference to the thesis of R. Achilles and the lectures close with illuminating examples, applications and some open questions.

MSC:

14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13-02 Research exposition (monographs, survey articles) pertaining to commutative algebra
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
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