Zariski surface
Encyclopedia
In algebraic geometry
, a branch of mathematics
, a Zariski surface is a surface
over a field
of characteristic p > 0 such that there is a dominant inseparable map of degree p from the projective plane
to the surface. In particular, all Zariski surfaces are unirational.
They were named by Piotr Blass after Oscar Zariski
who used them in 1958 to give examples of unirational surfaces in characteristic p > 0 that are not rational. (In characteristic 0 by contrast, Castelnuovo's theorem implies that all unirational surfaces are rational.)
Zariski surfaces are birational to surfaces in affine
3-space A3 defined by irreducible polynomial
s of the form
The following problem posed by Oscar Zariski in 1971 is still open: let p ≥ 5, let S be a Zariski surface with vanishing geometric genus. Is S necessarily a rational surface?
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Zariski surface is a surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
of characteristic p > 0 such that there is a dominant inseparable map of degree p from the projective plane
Projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...
to the surface. In particular, all Zariski surfaces are unirational.
They were named by Piotr Blass after Oscar Zariski
Oscar Zariski
Oscar Zariski was a Russian mathematician and one of the most influential algebraic geometers of the 20th century.-Education:...
who used them in 1958 to give examples of unirational surfaces in characteristic p > 0 that are not rational. (In characteristic 0 by contrast, Castelnuovo's theorem implies that all unirational surfaces are rational.)
Zariski surfaces are birational to surfaces in affine
Affine space
In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...
3-space A3 defined by irreducible polynomial
Irreducible polynomial
In mathematics, the adjective irreducible means that an object cannot be expressed as the product of two or more non-trivial factors in a given set. See also factorization....
s of the form
The following problem posed by Oscar Zariski in 1971 is still open: let p ≥ 5, let S be a Zariski surface with vanishing geometric genus. Is S necessarily a rational surface?