Negative curve on a projective surface ($C^2<0$)

A negative curve is an irreducible algebraic curve on a smooth projective surface whose self-intersection number is negative. The negativity prevents it from moving in a nontrivial effective linear system and constrains how many curves can represent a fixed numerical class.

Distinct negative curves cannot have the same integral cohomology class: their mutual intersection would equal a negative self-intersection, contradicting nonnegative intersection multiplicities for distinct curves. A smooth complex projective surface has finitely generated integral cohomology groups, so its negative curves form an at most countable set. Zero-self-intersection curves need not obey this restriction: one ruling of ${\mathbb{P}}^1\times {\mathbb{P}}^1$ gives an uncountable family.

If an effective divisor $D$ is linearly equivalent to a negative curve $C$, then $D=C$. Otherwise $D\cdot C=C^2<0$ forces $C$ to be a component of $D$, since distinct irreducible curves have nonnegative intersection multiplicities. Write $D=C+D^{\prime }$ with $D^{\prime }$ effective and linearly equivalent to zero. An ample divisor intersects $D^{\prime }$ positively unless $D^{\prime }=0$, while linear equivalence forces that intersection to vanish.