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A Remark on a Paper of Crachiola and Makar-Limanov

100%

Dryło R.

tom 59

nr 3

203-206

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 $𝔸¹_k$, 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.

On the stable equivalence problem for k[x,y]

100%

Dryło R.

Colloquium Mathematicum

2011

tom 124

nr 2

247-253

L. Makar-Limanov, P. van Rossum, V. Shpilrain and J.-T. Yu solved the stable equivalence problem for the polynomial ring k[x,y] when k is a field of characteristic 0. In this note we give an affirmative solution for an arbitrary field k.

Non-uniruledness and the cancellation problem (II)

100%

Dryło R.

nr 1

41-48

We study the following cancellation problem over an algebraically closed field 𝕂 of characteristic zero. Let X, Y be affine varieties such that $X × 𝕂^m ≅ Y × 𝕂^m$ for some m. Assume that X is non-uniruled at infinity. Does it follow that X ≅ Y? We prove a result implying the affirmative answer in case X is either unirational or an algebraic line bundle. However, the general answer is negative and we give as a counterexample some affine surfaces.

Non-uniruledness and the cancellation problem

100%

Dryło R.

2005

tom 87

nr 1

93-98

Using the notion of uniruledness we indicate a class of algebraic varieties which have a stronger version of the cancellation property. Moreover, we give an affirmative solution to the stable equivalence problem for non-uniruled hypersurfaces.