Double plane
An algebraic surface representing the two-dimensional analogue of a hyper-elliptic curve. A non-singular algebraic projective surface over an algebraically closed field
is said to be a double plane if its field of rational functions
is a quadratic extension of the field of rational functions in two variables. If the characteristic of the field is distinct from two (in what follows, this condition is assumed to hold), any double plane is birationally isomorphic to the affine surface given in three-dimensional affine space by an equation
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Surfaces of this type are sometimes also referred to as double planes. For each double plane there exists a morphism
into the projective plane
which splits into the composition of a birational morphism
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onto some normal surface and a finite morphism of the second degree
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The branching curve of the morphism
is said to be the branching curve of the double plane (and is not, in general, uniquely determined by
). The branching curve of a double plane plays an important role in the study of double planes. Thus, it may serve for the computation of numerical invariants of double planes. If
is irreducible, then the irregularity of the double plane
is zero. If the degree of
(which is always even) is
, and if all singularities of
are ordinary double or cuspidal points only (cf. Singular point of an algebraic curve), the arithmetic genus
and the Euler characteristic
(topological or
-adic) are calculated by the formulas:
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In the general case there exists a birational morphism such that the projection on
of the normalization
of the fibred product of
and
over
is a finite covering of degree 2 with a non-singular (and, possibly, reducible) branching curve
. In such case the following formulas are valid:
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For any curve of even degree on the projective plane there exists a double plane with
as its branching curve. The choice of a suitable curve
often makes it possible to solve the problem of the construction of an algebraic surface with given invariants [1], [3].
The classification of double planes is carried out in each class of algebraic surfaces separately. Rational and linear double planes have been described [5]; double planes which are elliptic surfaces or -surfaces have been listed [3] (cf. also Elliptic surface;
-surface). Numerous examples of double planes of fundamental type have been considered [3], .
References
[1] | I.V. Dolgachev, V.A. Iskovskikh, "Geometry of algebraic varieties" J. Soviet Math. , 5 : 6 (1976) pp. 803–864 Itogi Nauk. i Tekhn. Algebra Topol. Geom. , 12 (1974) pp. 77–170 |
[2] | O. Zariski, "Algebraic surfaces" , Springer (1971) MR0469915 Zbl 0219.14020 |
[3] | F. Enriques, "Le superficie algebraiche" , Bologna (1949) |
[4a] | L. Campedelli, "Siu piani doppi con curva di diramazione dell'ottavo ordine" Atti Accad. Naz. Lincei Rend., Ser. 6 , 15 (1932) pp. 203–208 |
[4b] | L. Campedelli, "Siu piani doppi con curva di diramazione del decimo ordine" Atti Accad. Naz. Lincei Rend., Ser. 6 , 15 (1932) pp. 358–362 |
[4c] | L. Campedelli, "Sopra alcuni piani doppi notevoli con curva di diramazione del decimo ordine" Atti Accad. Naz. Lincei Rend., Ser. 6 , 15 (1932) pp. 536–542 Zbl 0004.36306 Zbl 58.1232.02 |
[5] | H.W. Jung, "Rationale und halbrationale Doppelebene" J. Reine Angew. Math. , 184 : 4 (1942) pp. 199–237 |
Comments
References
[a1] | A. van de Ven, "Complex algebraic surfaces" , Springer (1984) |
Double plane. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Double_plane&oldid=24427