Rational surface
A two-dimensional algebraic variety, defined over an algebraically closed field , whose field of rational functions is a purely transcendental extension of
of degree 2. Every rational surface
is birationally isomorphic to the projective space
.
The geometric genus and the irregularity
of a complete smooth rational surface
are equal to 0, that is, there are no regular differential 2- or
-forms on
. Every multiple genus
of a smooth complete rational surface
is also zero, where
is the canonical divisor of the surface
. These birational invariants distinguish the rational surfaces among all algebraic surfaces, that is, any smooth complete algebraic surface with invariants
is a rational surface (the Castelnuovo rationality criterion). According to another rationality criterion, a smooth algebraic surface
is a rational surface if and only if there is a non-singular rational curve
on
with index of self-intersection
.
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 and the surfaces
(projectivization of two-dimensional vector bundles over the projective line
),
, where
and
. In other words, the surface
is a fibration by rational curves over a rational curve with a section
which is a smooth rational curve with index of self-intersection
. The surface
is isomorphic to the direct product
, and the surfaces
are obtained from
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 on a smooth complete rational surface is ample (cf. Ample sheaf), then
is called a Del Pezzo surface. The greatest integer
such that
for some divisor
on
is called the index of the Del Pezzo surface. The index
is equal to 1, 2 or 3 (see [2]). A Del Pezzo surface of index 3 is isomorphic to
. For a Del Pezzo surface
of index 2, the rational mapping
defined by the sheaf
gives a birational isomorphism onto a quadric in
. Del Pezzo surfaces of index 1 can be obtained by
monoidal transformations (cf. Monoidal transformation) of the plane
with centres at points in general position, where
(see [2]).
References
[1] | I.R. Shafarevich, "Algebraic surfaces" Proc. Steklov Inst. Math. , 75 (1967) Trudy Mat. Inst. Steklov. , 75 (1965) |
[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) |
Comments
If is defined over a, not necessarily algebraically closed, field and
is birationally equivalent to
over
, then
is said to be a
-rational surface.
References
[a1] | A. Beauville, "Surfaces algébriques complexes" Astérisque , 54 (1978) |
[a2] | J. Semple, L. Roth, "Introduction to algebraic geometry" , Oxford Univ. Press (1985) |
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