Part I. General formulae. In ordinary space of three dimensions, an algebraic surface is the locus represented by a single polynomial equation, F(x, y, z, t) = 0, homogeneous in the coordinates x, y, z, t. It is generally intended that the polynomial F is incapable of being written as the product of other polynomials, and the surface is then said to be irreducible. The order of the surface is the number of its intersections with an arbitrary line, and is the order of the polynomial F in x, y, z, t. Unless the contrary is stated we shall suppose the plane t = 0 to have no special relation with the surface, and shall often replace t by 1, and represent the equation by F(x, y, z) = 0.

An algebraic curve is most naturally regarded as the intersection of algebraic surfaces; two such surfaces meet in a curve. But it is not true conversely that any given algebraic curve is the complete intersection of two surfaces; a familiar example to the contrary is the rational cubic curve, which is the part intersection of two quadric surfaces having, beside, a line in common. Nor indeed is it clear that three algebraic surfaces can be drawn through a given curve so as to have this as their only common part; they may have, beside, points in common; in fact it will be seen below that a curve may be such that no three surfaces can be drawn through this which do not have also points in common not lying on this curve.