Singular Points and Tangent Cones

The Zariski tangent space to a variety X ⊂ ??n at a point p is described by taking the linear part of the expansion around p of all the functions on ??n 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 ⊂ ??2 is a plane curve, the Zariski tangent space to X at any singular point p will be all of T _p (??2) = K2. 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.