over C, and let p and q be distinct complex points of X. Then the line bundle is ample but has . Criteria for ampleness of line bundles Intersection theory To determine whether a given line bundle on a projective variety X is ample, the following numerical criteria (in terms of intersection numbers) are often the most useful. It is equivalent to ask when a Cartier divisor D on X is ample, meaning that the associated line bundle O(D) is ample. The intersection number can be defined as the degree of the line bundle O(D) restricted to C. In the other direction, for a line bundle L on a projective variety, the first Chern class means the associated Cartier divisor (defined up to linear equivalence), the divisor of any nonzero rational section of L. On a smooth projective curve X over an algebraically closed field k, a line bundle L is very ample if and only if for all k-rational points x,y in X. Let g be the genus of X. By the Riemann–Roch theorem, every line bundle of degree at least 2g + 1 satisfies this condition and hence is very ample. As a result, a line bundle on a curve is ample if and only if it has positive degree. For example, the canonical bundle of a curve X has degree 2g − 2, and so it is ample if and only if . The curves with ample canonical bundle form an important class; for example, over the complex numbers, these are the curves with a metric of negative curvature. The canonical bundle is very ample if and only if and the curve is not hyperelliptic. The Nakai–Moishezon criterion (named for Yoshikazu Nakai (1963) and Boris Moishezon (1964)) states that a line bundle L on a proper scheme X over a field is ample if and only if for every (irreducible) closed subvariety Y of X (Y is not allowed to be a point). In terms of divisors, a Cartier divisor D is ample if and only if for every (nonzero-dimensional) subvariety Y of X. For X a curve, this says that a divisor is ample if and only if it has positive degree. For X a surface, the criterion says that a divisor D is ample if and only if its self-intersection number is positive and every curve C on X has . Kleiman's criterion To state