On projective completions of affine varieties determined by 'degree-like' functions

TL;DR

This paper explores how to construct projective completions of affine varieties using degree-like functions, ensuring certain polynomial maps do not gain points at infinity, and characterizes these completions via algebraic properties.

Contribution

It introduces a method to create projective completions that preserve fibers of polynomial maps using finite collections of semidegrees, with a characterization based on radical ideals.

Findings

Existence of completions that do not add points at infinity for polynomial maps

Characterization of such completions via radical ideals of hypersurfaces at infinity

Establishment of an affine Bezout type theorem for specific polynomial maps

Abstract

We study projective completions of affine algebraic varieties which are given by filtrations, or equivalently, 'degree like functions' on their rings of regular functions. For a quasifinite polynomial map P (i.e. with all fibers finite) of affine varieties, we prove that there are completions of the source that do not add points at infinity for P (i.e. in the intersection of completions of the hypersurfaces corresponding to a generic fiber and determined by the component functions of P). Moreover we show that there are 'finite type' completions with the latter property, determined by the maximum of a finite number of 'semidegrees', i.e. maps of the ring of regular functions excluding zero, into integers, which send products into sums and sums into maximas (with a possible exception when the summands have the same semidegree). We characterize the latter type completions as the ones for which the ideal of the 'hypersurface at infinity' is radical. Moreover, we establish a one-to-one correspondence between the collection of minimal associated primes of the latter ideal and the unique minimal collection of semidegrees needed to define the corresponding degree like function. We also prove an 'affine Bezout type' theorem for quasifinite polynomial maps P which admit semidegrees such that corresponding completions do not add points at infinity for P.