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.

A cubic surface is a degree-three projective hypersurface in the projective space ${\mathbb{P}}^3$. Over an algebraically closed field every cubic surface contains a projective line. The smooth case has finitely many lines, although the incidence proof of existence does not require smoothness.

A cubic surface with empty singular locus. Over an algebraically closed field, it is irreducible: distinct positive-degree hypersurface components would meet, and every point of their intersection would be singular in their product equation.

For two skew lines $L,M$ on a smooth cubic surface, their joining line meets the surface at a third point generically. In coordinates with $L=\{x_2=x_3=0\}$ and $M=\{x_0=x_1=0\}$, write $F(\alpha A+\beta B)=\alpha \beta (\alpha U+\beta V)$. The third point is $[VA-UB]$ whenever $UV\ne 0$. Neither $U$ nor $V$ is identically zero, because that would make one of the two lines singular on the surface. The inverse records $[x_0:x_1]$ and $[x_2:x_3]$ of a general surface point. Thus the surface is birational to ${\mathbb{P}}^1\times {\mathbb{P}}^1$, hence to ${\mathbb{P}}^2$.

Write a line as $L=\{x_2=x_3=0\}$ and the surface equation as $F=x_2{Q}_2+x_3{Q}_3$. Smoothness says $A=Q_2{|}_L$, $B=Q_3{|}_L$ have no common zero. Projection $[x_2:x_3]$ extends over $L$ by $[-B:A]$. Its fibres are the residual plane conics in the planes through $L$, since substituting $x_2=sw$, $x_3=tw$ gives $F=w(s{Q}_2+t{Q}_3)$. Different residual conics are disjoint everywhere, including their points on $L$, because they are fibres of one morphism of algebraic varieties. The family is flat: over the smooth base curve its local rings are torsion-free over the corresponding discrete valuation ring; smoothness of the total surface supplies the reduced-fibre conclusions.

The matrix of the residual plane conic in coordinates $(u,v,w)$ has coefficient degrees $\left(\begin{array}{ccc}1& 1& 2\\ 1& 1& 2\\ 2& 2& 3\end{array}\right)$ in the plane parameter $(s,t)$. Every term of its determinant has degree five. The reduced singular fibres of a smooth conic bundle lemma rules out an identically zero determinant and makes each zero simple. Over an algebraically closed field, there are therefore exactly five singular fibres, each a pair of distinct coplanar projective lines. No residual conic contains the original line, because $sA+tB\equiv 0$ would make $A,B$ proportional and force a common zero. Pairs from different fibres are disjoint.

Every projective line through $P$ on a smooth cubic surface lies in its projective tangent plane. Each is a distinct linear factor of the cubic plane section. There can therefore be at most three. Equality occurs when the tangent section consists of three concurrent lines.

An exceptional curve of a surface resolution is an irreducible curve on the smooth resolving surface that maps to a point on the original normal surface. For an affine toric surface resolved by fan subdivision, the interior inserted rays give these proper curves. Each has a complete one-dimensional star and is isomorphic to the projective line.

A smooth algebraic surface is a two-dimensional smooth variety. Its local rings are regular, and its integral curves define Cartier divisors.

For an integral curve $C$ on a smooth projective surface, $2p_a(C)-2=C\cdot (C+K_X)$. Apply Riemann–Roch theorem for algebraic surfaces to the divisor restriction exact sequence $0\to {\mathcal{O}}_X(-C)\to {\mathcal{O}}_X\to {\mathcal{O}}_C\to 0$ to derive it. This arithmetic version applies also to singular curves and in arbitrary characteristic.

For an integral projective curve on a smooth projective surface, blowing up a point of multiplicity $m$ gives $C^{\prime }={\pi }^{\ast }C-mE$. The intersection formula for blowing up a surface and canonical divisor formula for a surface blowup yield $C^{\prime 2}=C^2-m^2$ and $K_{X^{\prime }}\cdot C^{\prime }=K_X\cdot C+m$. Substitute both in the arithmetic adjunction formula on a smooth surface. This does not require the transformed curve to be smooth.

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.

For a Cartier divisor on a smooth projective surface, the displayed formula computes its Euler characteristic of a coherent sheaf. Combined with Serre duality, it turns intersection numbers into lower bounds for section spaces.

For a smooth compact complex surface, the degree-four part of its Todd class is $(c_1^2+c_2)/12$. Applying Riemann–Roch to its structure sheaf gives the formula. For a K3 surface, $c_1=0$ and $\chi ({\mathcal{O}}_X)=2$, so its topological Euler characteristic is $c_2=24$.

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 a smooth projective surface and an ample divisor $A$, the intersection pairing on the Picard group of a surface is negative definite on the orthogonal complement of $[A]$ in $N^1(X{)}_{\mathbb{R}}$. Its signature is $(1,\rho -1)$. This is an algebraic statement valid in arbitrary characteristic, distinct from its analytic formulation for compact Kähler surfaces.

Let a surjective morphism from a smooth projective surface to a projective surface contract an irreducible curve $C$. Pull back an ample divisor $H$ from the target to obtain $L$. The projection formula for algebraic cycles gives $L\cdot C=0$ and $L^2=\mathrm{deg}(f)H^2>0$. The Hodge index theorem for algebraic surfaces makes the intersection form negative definite on the orthogonal complement of this positive-square class. Since an ample divisor on the source meets $C$ positively, its numerical class is nonzero, and $C^2<0$.

If $D$ is nonzero numerically with $D^2=0$ and $D\cdot A>0$ for ample $A$, then $D\cdot E=0$ implies $E^2\le 0$, with equality only when $[E]$ is proportional to $[D]$. To see this, use the Hodge index theorem for algebraic surfaces to split off the positive $A$ direction and diagonalize the remaining negative definite form. Orthogonality to a nonzero isotropic vector then leaves a negative semidefinite form whose radical is precisely that vector's span.

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.

This is precisely the curve contracted by the Castelnuovo contraction criterion.

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

Minimality here is the smooth-surface condition, rather than a claim about every higher-dimensional minimal model.

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.

Let $C$ be a smooth connected ample curve in an abelian surface $A$. Translate its chosen basepoint to zero and apply the universal property of the Jacobian variety. The image of $J(C)\to A$ is a closed connected algebraic subgroup containing the translated curve. If it had dimension one, that curve would be the subgroup itself; a distinct coset is disjoint and numerically equivalent, forcing $C^2=0$. This contradicts ampleness, so the homomorphism is surjective.

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.

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 $q(X)\le g$.