# 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 ).

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 ). 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 ).

How to Cite This Entry:
Rational surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rational_surface&oldid=52156
This article was adapted from an original article by Vik.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article