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. Classification of singularities Singular points include multiple points where the curve crosses over itself, and also various types of cusp, for example that shown by the curve with equation x3 = y2 at (0,0). A curve C has at most a finite number of singular points. If it has none, it can be called smooth or non-singular. Commonly, this definition is understood over an algebraically closed field and for a curve C in a projective space (i.e., complete in the sense of algebraic geometry). For example, the plane curve of equation is considered as singular, as having a singular point (a cusp) at infinity. In the remainder of this section, one considers a plane curve defined as the zero set of a bivariate polynomial . Some of the results, but not all, may be generalized to non-plane curves. The singular points are classified by means of several invariants. The multiplicity is defined as the maximum integer such that the derivatives of to all orders up to vanish (also the minimal intersection number between the curve and a straight line at ). Intuitively, a singular point has if it concentrates ordinary double points at . To make this precise, the blow up process produces so-called infinitely near points, and summing over the infinitely near points, where m is their multiplicity, produces . For an irreducible and reduced curve and a point we can define algebraically as the length of where is the local ring at P and is its integral closure. The Milnor number of a singularity is the degree of the mapping on the small sphere of radius ε, in the sense of the topological degree of a continuous mapping, where is the (complex) gradient vector field of f. It is related to δ and r by the Milnor–Jung formula, Here, the branching number r of P is the number of locally irreducible branches at P. For example, r = 1 at an ordinary cusp, and r = 2 at an ordinary double point. The multiplicity