Rational surface

A two-dimensional algebraic variety, defined over an algebraically closed field $k$, whose field of rational functions is a purely transcendental extension of $k$ of degree 2. Every rational surface $X$ is birationally isomorphic to the projective space $\mathbf P ^ {2}$.

The geometric genus $p _ {g}$ and the irregularity $q$ of a complete smooth rational surface $X$ are equal to 0, that is, there are no regular differential 2- or 1-forms on $X$. Every multiple genus $P _ {n} = \mathop{\rm dim} H ^ {0} ( X , {\mathcal O} _ {X} ( n K _ {X} ) )$ of a smooth complete rational surface $X$ is also zero, where $K _ {X}$ is the canonical divisor of the surface $X$. These birational invariants distinguish the rational surfaces among all algebraic surfaces, that is, any smooth complete algebraic surface with invariants $p _ {g} = q = P _ {2} = 0$ is a rational surface (the Castelnuovo rationality criterion). According to another rationality criterion, a smooth algebraic surface $X$ is a rational surface if and only if there is a non-singular rational curve $C$ on $X$ with index of self-intersection $( C ^ {2} ) _ {X} > 0$.

With the exception of rational surfaces and ruled surfaces, every algebraic surface is birationally isomorphic to a unique minimal model. In the class of rational surfaces there is a countable set of minimal models. It consists of the projective space $\mathbf P ^ {2}$ and the surfaces $F _ {n} \simeq P ( {\mathcal L} _ {n} )$ (projectivization of two-dimensional vector bundles over the projective line $\mathbf P ^ {1}$), ${\mathcal L} _ {n} \simeq {\mathcal O} _ {\mathbf P ^ {1} } \oplus {\mathcal O} _ {\mathbf P ^ {1} } ( - n )$, where $n \geq 0$ and $n \neq 1$. In other words, the surface $F _ {n}$ is a fibration by rational curves over a rational curve with a section $S _ {n}$ which is a smooth rational curve with index of self-intersection $( S _ {n} ^ {2} ) _ {F} = - n$. The surface $F _ {0}$ is isomorphic to the direct product $\mathbf P ^ {1} \times \mathbf P ^ {1}$, and the surfaces $F _ {n}$ are obtained from $F _ {0}$ by a sequence of elementary transformations (see [1]).

Rational surfaces have a large group of birational transformations (called the group of Cremona transformations).

If the anti-canonical sheaf ${\mathcal O} _ {X} ( - K _ {X} )$ on a smooth complete rational surface is ample (cf. Ample sheaf), then $X$ is called a Del Pezzo surface. The greatest integer $r > 0$ such that $- K _ {X} \sim r D$ for some divisor $D$ on $X$ is called the index of the Del Pezzo surface. The index $r$ is equal to 1, 2 or 3 (see [2]). A Del Pezzo surface of index 3 is isomorphic to $\mathbf P ^ {2}$. For a Del Pezzo surface $X$ of index 2, the rational mapping ${\mathcal O} _ {X} : X \rightarrow \mathbf P ^ {3}$ defined by the sheaf ${\mathcal O} _ {X} ( D)$ gives a birational isomorphism onto a quadric in $\mathbf P ^ {3}$. Del Pezzo surfaces of index 1 can be obtained by $n$ monoidal transformations (cf. Monoidal transformation) of the plane $\mathbf P ^ {2}$ with centres at points in general position, where $1 \leq n \leq 8$ (see [2]).

References

[1]	I.R. Shafarevich, "Algebraic surfaces" Proc. Steklov Inst. Math. , 75 (1967) Trudy Mat. Inst. Steklov. , 75 (1965) MR1392959 MR1060325 Zbl 0830.00008 Zbl 0733.14015 Zbl 0832.14026 Zbl 0509.14036 Zbl 0492.14024 Zbl 0379.14006 Zbl 0253.14006 Zbl 0154.21001
[2]	V.A. Iskovskii, "Anticanonical models of three-dimensional algebraic varieties" , Current problems in mathematics , 12 , Moscow (1979) pp. 59–157; 239 (In Russian)
[3]	R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001

Comments

If $X$ is defined over a, not necessarily algebraically closed, field and $X$ is birationally equivalent to $\mathbf P _ {k} ^ {2}$ over $k$, then $X$ is said to be a $k$-rational surface.

References

[a1]	A. Beauville, "Surfaces algébriques complexes" Astérisque , 54 (1978) MR0485887 Zbl 0394.14014
[a2]	J. Semple, L. Roth, "Introduction to algebraic geometry" , Oxford Univ. Press (1985) MR0814690 Zbl 0576.14001