Smooth projective surface

A smooth projective surface is a two-dimensional smooth variety admitting a closed embedding in projective space. Its integral curves are Cartier divisors, and their intersections, arithmetic adjunction and point blowups control its birational geometry.

A very general smooth degree-$d$ surface $X\subset {\mathbb{P}}_{\mathbb{C}}^3$ with $d\ge 4$ has Picard group generated by its hyperplane class. The exceptional surfaces lie in a countable union of proper closed loci. A proof is given by Griffiths and Harris, "On the Noether-Lefschetz Theorem".

The Noether–Lefschetz theorem makes any curve class a positive multiple of the hyperplane class, with positive self-intersection. But the smooth rational-curve self-intersection on a projective hypersurface formula makes every smooth rational curve have self-intersection at most $-2$ when the surface degree is at least four. Thus none exists. The smoothness condition is essential; this does not prohibit singular curves whose normalizations are rational.

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.

A smooth rational curve with self-intersection $-1$ on a smooth projective surface over an algebraically closed field contracts to a smooth point of another smooth projective surface. The contraction is a birational morphism and an isomorphism off the curve, with inverse a blowup of a smooth algebraic surface. This algebraic criterion is valid in arbitrary characteristic and should not be confused with Castelnuovo's rationality criterion.