structure of an abelian group with the distinguished point as the identity of the group law. In a plane cubic model three points sum to zero in the group if and only if they are collinear. For an elliptic curve defined over the complex numbers the group is isomorphic to the additive group of the complex plane modulo the period lattice of the corresponding elliptic functions. The intersection of two quadric surfaces is, in general, a nonsingular curve of genus one and degree four, and thus an elliptic curve, if it has a rational point. In special cases, the intersection either may be a rational singular quartic or is decomposed in curves of smaller degrees which are not always distinct (either a cubic curve and a line, or two conics, or a conic and two lines, or four lines). Curves of genus greater than one Curves of genus greater than one differ markedly from both rational and elliptic curves. Such curves defined over the rational numbers, by Faltings's theorem, can have only a finite number of rational points, and they may be viewed as having a hyperbolic geometry structure. Examples are the hyperelliptic curves, the Klein quartic curve, and the Fermat curve when is greater than three. Also projective plane curves in and curves in provide many useful examples. Projective plane curves Plane curves of degree , which can be constructed as the vanishing locus of a generic section , has genus which can be computed using Coherent sheaf cohomology. Here's a brief summary of the curves genera relative to their degree For example, the curve defines a curve of genus which is smooth since the differentials have no common zeros with the curve.. A non-example of a generic section is the curve which, by Bezouts theorem, should intersect at most points, is the union of two rational curves intersecting at two points. Note is given by the vanishing locus of and is given by the vanishing locus of . These can be found explicitly: a point lies in both if . So the two solutions are the points such that , which are and . Curves in product of projective lines Curve given by the vanishing locus of , for , give curves of genuswhich can be checked using Coherent sheaf cohomology. If , then they define curves of genus , hence a curve of any genus can be