The third neighbourhood of a point in a plane

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An unexceptionally birational model is constructed, in the form of a non singular algebraic threefold variety W, of the three dimensional aggregate of plane curve elements P0P1P2P3 of the third rank with origin at a fixed point P0 of the plane. The geometric properties of W are then examined, and a base for surfaces and curves on W found. In particular the mapping on W of the singular (for instance cuspidal) curve elements is clarified.

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To Enrico Bompiani on his scientific Jubilee

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Du Val, P. The third neighbourhood of a point in a plane. Annali di Matematica 53, 301–314 (1961).

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  • Geometric Property
  • Plane Curve
  • Curve Element
  • Birational Model
  • Dimensional Aggregate