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1. CMB Online first
Homological Planes in the Grothendieck Ring of Varieties In this note, we identify, in the Grothendieck group of complex
varieties $K_0(\mathrm Var_\mathbf{C})$, the classes of $\mathbf{Q}$-homological
planes. Precisely, we prove that a connected smooth affine complex
algebraic surface $X$ is a $\mathbf{Q}$-homological plane if
and only if $[X]=[\mathbf{A}^2_\mathbf{C}]$ in the ring $K_0(\mathrm Var_\mathbf{C})$
and $\mathrm{Pic}(X)_\mathbf{Q}:=\mathrm{Pic}(X)\otimes_\mathbf{Z}\mathbf{Q}=0$.
Keywords:motivic nearby cycles, motivic Milnor fiber, nearby motives Categories:14E05, 14R10 |
2. CMB 2009 (vol 52 pp. 535)
A Note on Locally Nilpotent Derivations\\ and Variables of $k[X,Y,Z]$ We strengthen certain results
concerning actions of $(\Comp,+)$ on $\Comp^{3}$
and embeddings of $\Comp^{2}$ in $\Comp^{3}$,
and show that these results are in fact valid
over any field of characteristic zero.
Keywords:locally nilpotent derivations, group actions, polynomial automorphisms, variable, affine space Categories:14R10, 14R20, 14R25, 13N15 |
3. CMB 2005 (vol 48 pp. 622)
Hyperplanes of the Form ${f_1(x,y)z_1+\dots+f_k(x,y)z_k+g(x,y)}$ Are Variables The Abhyankar--Sathaye Embedded Hyperplane Problem asks whe\-ther any
hypersurface of $\C^n$ isomorphic to $\C^{n-1}$ is rectifiable, {\em
i.e.,}
equivalent to a linear hyperplane up to an automorphism of $\C^n$.
Generalizing the approach adopted by Kaliman, V\'en\'ereau, and
Zaidenberg which
consists in using almost nothing but the acyclicity of $\C^{n-1}$, we solve
this problem for hypersurfaces given by polynomials of $\C[x,y,z_1,\dots, z_k]$
as in the title.
Keywords:variables, Abhyankar--Sathaye Embedding Problem Categories:14R10, 14R25 |