Primitive Normal completions of the affine plane I

Details Primitive Normal completions of the affine plane I Primitive Normal completions of the affine plane I

2011-10-31

arxiv.org/abs/1110.6905v1

Mathematics

Algebraic Geometry

Scientific paper

In this article we study normal compactifications of $\cc^2$ from the point of view of (discrete) valuations associated to the curves at infinity, or equivalently, pencils of `jets of curve-germs' centered at points at infinity. We give an explicit (and easy to calculate) characterization of discrete valuations which correspond to normal compactifications of $\cc^2$ which are {\em primitive} (i.e.\ the curve at infinity is irreducible). We also calculate several invariants of the primitive compactifications of $\cc^2$ in terms of the corresponding curve-jets. As a consequence of these calculations we derive a new proof of Jung's theorem on polynomial automorphisms of $\cc^2$.

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