In the paper [4] Krasiński and Spodzieja proved that if f : X -> Y is a Zariski closed non-constant mapping of affine varieties over C (where dim X [is greater than or equal to] 2), then f is finite. In this paper we generalize this result to the case of arbitrary algebraically closed field.
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Let X be a smooth affine variety of dimension n > 2. Assume that the group H[sub 1](X,Z) is a torsion group and that [chi](X) = 1. Let Y be a projectively smooth affine hypersurface Y [is a subset of] C[sup n+1] of degree d > 1, which is smooth at infinity. Then there is no injective polynomial mapping f : X --> Y. This contradicts a result of Peretz [5].
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We construct general examples of affine normal varieties X which have a hypersurface H [is a subset of] X, such that the variety X [...] H is not affine. We also show, that if X is an affine normal surface and H is a curve in X, then X [...] H is affine, too.
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The aim of this paper is to give a simple method of computing the set S[f] of points at which a generically-finite polynomial mapping f : [C^2 --> C^2] is not proper.
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Let K be an algebraically closed field. For every n>2 we find an n-dimensional complete variety Xn over K, which has infinitely many maximal quasi-projecive open subsets.
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