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Homological Planes in the Grothendieck Ring of Varieties
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- Journal:
- Canadian Mathematical Bulletin / Volume 58 / Issue 2 / 01 June 2015
- Published online by Cambridge University Press:
- 20 November 2018, pp. 356-362
- Print publication:
- 01 June 2015
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- Article
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In this note we identify the classes of
$\text{Q}$-homological planes in the Grothendieck group of complex varieties 
${{K}_{0}}\left( \text{Va}{{\text{r}}_{\text{C}}} \right)$. Precisely, we prove that a connected, smooth, affine, complex, algebraic surface 
$X$ is a $\text{Q}$-homological plane if and only if 
$\left[ X \right]\,=\,\left[ \text{A}_{\text{C}}^{2} \right]$ in the ring ${{K}_{0}}\left( \text{Va}{{\text{r}}_{\text{C}}} \right)$ and 
$\text{Pic}{{\left( X \right)}_{\text{Q}}}\,:=\,\text{Pic}\left( X \right)\,{{\otimes }_{\text{Z}}}\,\text{Q}\,\text{=}\,\text{0}$.
