Singular Points and Tangent Cones

  • Joe Harris
Part of the Graduate Texts in Mathematics book series (GTM, volume 133)


The Zariski tangent space to a variety X ⊂ 𝔸n at a point p is described by taking the linear part of the expansion around p of all the functions on 𝔸n vanishing on X. In case p is a singular point of X, however, this does not give us a very refined picture of the local geometry of X; for example, if X ⊂ 𝔸2 is a plane curve, the Zariski tangent space to X at any singular point p will be all of T p (𝔸 2) = K 2. We will describe here the tangent cone, an object that, while it certainly does not give a complete description of the local structure of a variety at a singular point, is at least a partial refinement of the notion of tangent space.


Singular Point Tangent Space Tangent Line Plane Curve Double Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Joe Harris
    • 1
  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA

Personalised recommendations