algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, a branch of

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a Zariski surface is a

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...

over a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

of characteristic ''p'' > 0 such that there is a dominant inseparable map of degree ''p'' from the

projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, paral ...

to the surface. In particular, all Zariski surfaces are

unirational In mathematics, a rational variety is an algebraic variety, over a given field ''K'', which is birationally equivalent to a projective space of some dimension over ''K''. This means that its function field is isomorphic to :K(U_1, \dots , U_d), t ...

. They were named by Piotr Blass in 1977 after

Oscar Zariski Oscar Zariski (April 24, 1899 – July 4, 1986) was an American mathematician. The Russian-born scientist was one of the most influential algebraic geometers of the 20th century. Education Zariski was born Oscher (also transliterated as Ascher o ...

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 In algebraic geometry, a branch of mathematics, a rational surface is a surface birationally equivalent to the projective plane, or in other words a rational variety of dimension two. Rational surfaces are the simplest of the 10 or so classes of su ...

implies that all unirational surfaces are rational.) Zariski surfaces are

birational In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational f ...

to surfaces in

affine Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a Affinity_(law)#Terminology, relative by marriage in law and anthropology * Affine cipher, a special case of the more general substi ...

3-space ''A''³ defined by

irreducible polynomial In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted f ...

s of the form :

z^p = f(x, y).\

The following problem was posed by Oscar Zariski in 1971: Let ''S'' be a Zariski surface with vanishing geometric

genus Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...

. Is S necessarily a rational surface? For ''p'' = 2 and for ''p'' = 3 the answer to the above problem is negative as shown in 1977 by Piotr Blass in his

University of Michigan The University of Michigan (U-M, U of M, or Michigan) is a public university, public research university in Ann Arbor, Michigan, United States. Founded in 1817, it is the oldest institution of higher education in the state. The University of Mi ...

Ph.D. thesis and by William E. Lang in his Harvard Ph.D. thesis in 1978. announced further examples giving a negative answer to Zariski's question in every characteristic p>0 . His method however is non constructive at the moment and we do not have explicit equations for p>3.

References

* * *{{Citation , last1=Zariski , first1=Oscar , author1-link=Oscar Zariski , title=On Castelnuovo's criterion of rationality ''p''_''a''=''P''₂=0 of an algebraic surface , url=http://projecteuclid.org/euclid.ijm/1255454536 , mr=0099990 , year=1958 , journal=Illinois Journal of Mathematics , issn=0019-2082 , volume=2 , pages=303–315 Algebraic surfaces University of Michigan

See also

References