A characterization of proper regular mappings
Let X, Y be complex affine varieties and f:X → Y a regular mapping. We prove that if dim X ≥ 2 and f is closed in the Zariski topology then f is proper in the classical topology.
A class of counterexamples to the Cancellation Problem for arbitrary rings
We present a class of counterexamples to the Cancellation Problem over arbitrary commutative rings, using non-free stably free modules and locally nilpotent derivations.
A Note on Elementary Derivations
2000 Mathematics Subject Classification: Primary: 14R10. Secondary: 14R20, 13N15.Let R be a UFD containing a field of characteristic 0, and Bm = R[Y1, . . . , Ym] be a polynomial ring over R. It was conjectured in [5] that if D is an R-elementary monomial derivation of B3 such that ker D is a finitely generated R-algebra then the generators of ker D can be chosen to be linear in the Yi ’s. In this paper, we prove that this does not hold for B4. We also investigate R-elementary derivations D of Bm...
A Note on Geometric Degree of Finite Extensions of Mappings from a Smooth Variety
A note on triangular automorphisms.
A Remark on a Paper of Crachiola and Makar-Limanov
A. Crachiola and L. Makar-Limanov [J. Algebra 284 (2005)] showed the following: if X is an affine curve which is not isomorphic to the affine line , then ML(X×Y) = k[X]⊗ ML(Y) for every affine variety Y, where k is an algebraically closed field. In this note we give a simple geometric proof of a more general fact that this property holds for every affine variety X whose set of regular points is not k-uniruled.
Abhyankar-Moh property and unique affine embeddings.
About induced orthogonality and generalized reflections of affine coordinate plane of odd order.
Affine rulings of weighted projective planes
It is explained that the following two problems are equivalent: (i) describing all affine rulings of any given weighted projective plane; (ii) describing all weighted-homogeneous locally nilpotent derivations of k[X,Y,Z]. Then the solution of (i) is sketched. (Outline of our joint work with Peter Russell.)
An application of algebraic geometry to encryption: tame transformation method.