= Countability of negative curves
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.
Back to article page
