Algebraic surface

An algebraic surface is a two-dimensional algebraic variety. The birational classification of smooth projective surfaces uses their curves, divisor intersections and pluricanonical maps.

On a smooth projective surface, two line bundles are represented by divisors $D$ and $E$, and their intersection is the degree of the zero-cycle obtained after moving them into proper position. It is a symmetric bilinear form on the Picard group, and $D\cdot E$ equals the degree of ${\mathcal{O}}_X(D)$ restricted to $E$ when $E$ is an integral curve not contained in $D$.

For the Hirzebruch surface ${\mathbb{F}}_n$, the negative section of a Hirzebruch surface $S$ and a fiber class of a Hirzebruch surface $F$ freely generate $\mathrm{Pic}({\mathbb{F}}_n)$ and satisfy

If $\pi :\tilde{X}\to X$ blows up a smooth point with exceptional curve $E$, thenIf a curve $C$ has multiplicity $m$ at the center, its strict transform is $\tilde{C}={\pi }^{\ast }C-mE$ and ${\tilde{C}}^2=C^2-m^2$.

Every nonisomorphic birational morphism from a smooth projective surface to ${\mathbb{P}}^2$ factors into point blowups. If $H$ is the pullback of a line and $E$ is the total transform of an exceptional divisor from one factor, then $H^2=1$, $E^2=-1$ and $H\cdot E=0$, so $(H+E{)}^2=0$.

Blowing up $r$ distinct smooth points of a smooth curve $C$ gives a strict transform $C^{\prime }$ isomorphic to $C$ withChoosing $r>C^2$ turns a curve of positive self-intersection into one of negative self-intersection without changing its abstract isomorphism type.

Every morphism $f:{\mathbb{P}}^2\to {\mathbb{P}}^n$ satisfies $f^{\ast }\mathcal{O}(1)\cong \mathcal{O}(d)$ for some $d\ge 0$. If $f$ contracts a line, restriction to that line makes $\mathcal{O}(d)$ trivial, hence $d=0$ and all defining sections are constant. Thus the entire morphism is constant.

An exceptional curve of the first kind is a smooth rational curve $E$ on a smooth surface with $E^2=-1$. It can be contracted to a smooth point.

A smooth projective surface is minimal when it contains no exceptional curve of the first kind.

An abelian surface is a two-dimensional abelian variety. It contains no rational curve: a morphism from ${\mathbb{P}}^1$ to an abelian variety is constant. Consequently every abelian surface is a minimal algebraic surface.

The Hirzebruch surface ${\mathbb{F}}_n$ is minimal for $n=0$ and every $n\ge 2$; ${\mathbb{F}}_1$ is not minimal because its negative section is a $(-1)$-curve. For $n>0$, the negative section is the unique irreducible curve of negative self-intersection, so its self-intersection $-n$ distinguishes the isomorphism class. The surfaces ${\mathbb{F}}_n$ for $n\ge 2$ therefore give infinitely many pairwise nonisomorphic minimal rational surfaces.

A K3 surface is a smooth projective surface with trivial canonical bundle and $H^1(X,{\mathcal{O}}_X)=0$. Over the complex numbers it is equivalently a simply connected compact complex surface with a nowhere-vanishing holomorphic two-form.

If a smooth curve $C$ of genus $g$ lies on a K3 surface, the Adjunction formula and $K_X=0$ give

A smooth quartic surface $X\subseteq {\mathbb{P}}^3$ containing a line $L$ is a K3 surface. The pencil of planes through $L$ cuts out $L$ plus a residual plane cubic; the divisor class $H-L$ has square zero and its basepoint-free pencil defines an elliptic fibration $X\to {\mathbb{P}}^1$.

An elliptic surface is a smooth projective surface equipped with a surjective morphism to a smooth curve whose generic fiber is a smooth curve of genus one.

For an elliptic curve $E$, projection ${\mathbb{P}}^1\times E\to {\mathbb{P}}^1$ is an elliptic fibration. Its canonical bundle is the pullback of $K_{{\mathbb{P}}^1}$, so it has no nonzero pluricanonical sections and its Kodaira dimension is $-\infty$.

The Kodaira dimension measures the asymptotic growth of the pluricanonical spaces $H^0(X,K_X^{\otimes m})$. For a surface it takes values $-\infty ,0,1,2$.

A smooth projective surface is of general type when its Kodaira dimension is two, equivalently when its canonical divisor is big.

Let $B\subseteq {\mathbb{P}}^2$ be a smooth curve of degree $2d$, and let $\pi :X\to {\mathbb{P}}^2$ be the double cover branched along $B$. ThenFor $d\ge 4$ this canonical divisor is ample, so $X$ is a surface of general type. A smooth octic branch curve gives the first such example.

The irregularity of a smooth projective complex surface is

For smooth projective curves $C$ and $D$, the Künneth theorem gives

A smooth surface hypersurface $X_d\subseteq {\mathbb{P}}^3$ has $q(X_d)=0$. This follows from the structure-sheaf sequence of a hypersurface and the vanishing of the intermediate cohomology of line bundles on ${\mathbb{P}}^3$.

If $C$ and $D$ have positive genus, then $q(C\times D)=g(C)+g(D)>0$, whereas every smooth surface hypersurface in projective space lies in ${\mathbb{P}}^3$ and has irregularity zero. Hence $C\times D$ is not isomorphic to a projective hypersurface.

The geometric genus of a smooth projective surface is

If $\pi :\tilde{X}\to X$ is the blowup of a smooth surface at a point, then $K_{\tilde{X}}={\pi }^{\ast }{K}_X+E$ and ${\pi }_{\ast }{K}_{\tilde{X}}=K_X$. Pullback therefore identifies the global holomorphic two-forms and gives $p_g(\tilde{X})=p_g(X)$.

The Albanese variety of a smooth projective variety $X$ is the universal abelian variety receiving a morphism from $X$ after a base point is chosen. Over the complex numbers its dimension is $q(X)$.

If the image of the Albanese morphism of a surface is a smooth curve $C$ of genus $g$, then $q(X)=g$. Pullback of one-forms from $C$ gives $g\le q(X)$, while the universal property gives a surjection from the Jacobian of $C$ onto $\mathrm{Alb}(X)$ and hence $q(X)\le g$.