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Please use this identifier to cite or link to this item: http://hdl.handle.net/1807/24371

Title: Towards a Bezout-type Theory of Affine Varieties
Authors: Mondal, Pinaki
Advisor: Milman, Pierre
Department: Mathematics
Keywords: Affine varieties
Projective completion
Bezout theorem
Toric varieties
Degree like function
Filtration
Semidegree
Issue Date: 21-Apr-2010
Abstract: We study projective completions of affine algebraic varieties (defined over an algebraically closed field K) which are given by filtrations, or equivalently, integer valued `degree like functions' on their rings of regular functions. For a polynomial map P := (P_1, ..., P_n): X -> K^n of affine varieties with generically finite fibers, we prove that there are completions of the source such that the intersection of completions of the hypersurfaces {P_j = a_j} for generic (a_1, ..., a_n) in K^n coincides with the respective fiber (in short, the completions `do not add points at infinity' for P). Moreover, we show that there are `finite type' completions with the latter property, i.e. determined by the maximum of a finite number of `semidegrees', by which we mean degree like functions that send products into sums. We characterize the latter type completions as the ones for which ideal I of the `hypersurface at infinity' is radical. Moreover, we establish a one-to-one correspondence between the collection of minimal associated primes of I and the unique minimal collection of semidegrees needed to define the corresponding degree like function. We also prove an `affine Bezout type' theorem for polynomial maps P with finite fibers that admit semidegrees corresponding to completions that do not add points at infinity for P. For a wide class of semidegrees of a `constructive nature' our Bezout-type bound is explicit and sharp.
URI: http://hdl.handle.net/1807/24371
Appears in Collections:Doctoral
Department of Mathematics - Doctoral theses

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