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Use countability of negative curves. A smooth projective surface over $\mathbb{C}$ is a compact smooth manifold homotopy equivalent to a finite CW complex, so its integral cohomology group $H^2(X,\mathbb{Z})$ is finitely generated and therefore countable. Distinct irreducible curves $C,D$ with $C^2<0$ cannot have the same cohomology class: that would give $C\cdot D=C^2<0$, while distinct effective curves have nonnegative intersection multiplicities. Assigning a negative curve its cohomology class is thus injective, proving there are at most countably many negative curves.

For zero self-intersection, uncountable families do occur. On the product of two projective lines ${\mathbb{P}}^1\times {\mathbb{P}}^1$, every fiber ${\mathbb{P}}^1\times \{p\}$ has self-intersection zero, since a different fiber is disjoint and has the same numerical equivalence of divisors class. The parameter $p\in {\mathbb{P}}^1(\mathbb{C})$ varies over an uncountable set. Thus yes, there can be uncountably many such curves.

The Nakai–Moishezon criterion says that a Cartier divisor $D$ on a projective scheme is ample exactly whenIn particular the test includes each positive-dimensional irreducible component. Here and below, “proper” means proper over the ground field, not necessarily a strict subset of $X$. Interpreting it as a strict subset would make the later criteria false even for an integral projective curve, whose strict closed subvarieties have dimension zero.

The intersection product of Cartier divisors with cycles depends only on their numerical equivalence of divisors classes. One way to see this is to intersect all but one factor first, obtaining a one-cycle; replacing the remaining factor by a numerically equivalent divisor does not change its pairing with that cycle. Multilinearity then handles replacement of every factor. Consequently $D\equiv D^{\prime }$ makes all the displayed numbers equal. ThusThe criterion applies to the integral reduced subvarieties of a possibly nonreduced scheme; ampleness on reduced components explains why the nilpotent structure does not change this condition.

Condition (b) immediately implies (c): a positive real multiple $tA$ of an ample divisor is an ample real divisor, and real linear equivalence of divisors implies numerical equivalence of divisors.

More explicitly, the real numerical class of $D$ is the sum of a class in the ample cone and an effective real divisor class. This separates strict positivity from the possibly degenerate effective part; it does not claim that the effective part itself is ample.

Real linear equivalence means that $D-E$ is a finite real combination of principal Cartier divisors. Rational linear equivalence uses rational coefficients instead. Both imply numerical equivalence of divisors.

The finite-dimensional real vector space $N^1(X{)}_{\mathbb{R}}$ consists of real combinations of Cartier divisors modulo numerical equivalence of divisors. Its elements are numerical divisor classes. The space of curve classes paired with it is $N_1(X{)}_{\mathbb{R}}$.