Plain Varieties
G\'abor Bodn\'ar, Herwig Hauser, Josef Schicho, Orlando Villamayor
TL;DR
Plain varieties are a class of algebraic varieties that are locally isomorphic to open subsets of affine space, known to be smooth and rational, with stability under blowups in smooth centers, but their characterization in higher dimensions remains open.
Contribution
The paper introduces the concept of plain varieties, proves their stability under blowups in smooth centers, and discusses their properties and open questions in higher dimensions.
Findings
Plain varieties are smooth and rational.
They are stable under blowup in smooth centers.
Characterization in higher dimensions remains open.
Abstract
Algebraic varieties which are locally isomorphic to open subsets of affine space will be called {\em plain}. Plain varieties are smooth and rational. The converse is true for curves and surfaces, and unknown in general. It is shown that plain varieties are stable under blowup in smooth centers.
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