Algebraic Curve - Singularities

Singularities

Using the intrinsic concept of tangent space, points P on an algebraic curve C are classified as smooth or non-singular, or else singular. Given n−1 homogeneous polynomials in n+1 variables, we may find the Jacobian matrix as the (n−1)×(n+1) matrix of the partial derivatives. If the rank of this matrix is n−1, then the polynomials define an algebraic curve (otherwise they define an algebraic variety of higher dimension). If the rank remains n−1 when the Jacobian matrix is evaluated at a point P on the curve, then the point is a smooth or regular point; otherwise it is a singular point. In particular, if the curve is a plane projective algebraic curve, defined by a single homogeneous polynomial equation f(x,y,z) = 0, then the singular points are precisely the points P where the rank of the 1×(n+1) matrix is zero, that is, where

Since f is a polynomial, this definition is purely algebraic and makes no assumption about the nature of the field F, which in particular need not be the real or complex numbers. It should of course be recalled that (0,0,0) is not a point of the curve and hence not a singular point.

Similarly, for an affine algebraic curve defined by a single polynomial equation f(x,y) = 0, then the singular points are precisely the points P of the curve where the rank of the 1×n Jacobian matrix is zero, that is, where

The singularities of a curve are not birational invariants. However, locating and classifying the singularities of a curve is one way of computing the genus, which is a birational invariant. For this to work, we should consider the curve projectively and require F to be algebraically closed, so that all the singularities which belong to the curve are considered.