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In algebraic geometry, a branch of mathematics, a Zariski surface is a surface over a field of characteristic p > 0 such 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 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 A defined by irreducible polynomials of the form

z = f(x, y).

Properties of Zariski surfaces

The Picard group of the generic Zariski surface has been computed using some ideas of Pierre Deligne and Alexander Grothendieck during 1980-1993.

Any Zariski surface with vanishing bigenus is rational and that all Zariski surfaces are simply connected. Zariski surfaces form a rich family including surfaces of general type, K3 surfaces, Enriques surfaces, quasi elliptic surfaces and also rational surfaces. In every characteristic the family of birationally distinct Zariski surfaces is infinite.

K3 surfaces

The Japanese school of geometers, following work of the Russian school of geometers, recently showed that every supersingular K3 surface in characteristic two is a Zariski surface. Michael Artin had previously invented a subtle numerical invariant called the Artin invariant that gives an important stratification of the moduli space of such supersingular K3 surfaces in characteristic two. The Albanese variety of a Zariski surface is always trivial. However as was shown by David Mumford school of geometry the Picard scheme need not be reduced again we refer to William E.Lang Harvard 1978 Ph.D. thesis. In 1980 Spencer Bloch and Piotr Blass proved that a Zariski surface which is irrational does not admit a finite purely inseparable degree p map onto the projective plane. Iacopo Barsotti remarked that this illustrates a very strong form of simple connectivity of the projective plane.

Bertini's theorem

Zariski surfaces illustrate a modification of the theorem of Bertini in characteristic p>0. Thus this above research about Zariski surfaces naturally led to exploring theorems of Bertini type.Alexander Grothendieck allowed his notes on Bertini type theorems to be published under the title EGA 5; they are available translated and partially edited at the website of the Grothendieck circle centered in France and at the website of James Milne from the University of Michigan.

Several articles about Zariski surfaces and Grothedieck's EGA 5 papers appeared in the Ulam Quarterly Journal.

Zariski B surfaces

For any smooth projective surface B in characteristic p>0 we shall call a Zariski B surface any smooth projective surface S whose function field arises from the function field of B by adjoining a p-th root. When B is an abelian surface Barsotti and Mumford have developed examples and partial theory prior to 1980. Another interesting and perhaps quite realistic case is when B is a surface of general type with Picard group isomorphic to the integers and generated by a hyperplane section in a suitable embedding of B in a projective space. Singularities that arise in the theory of Zariski surfaces and Zariski B surfaces have been resolved by Abyankhar,Zariski and Lipmann during the period 1956-1980. These singularities can be very complicated and difficult to resolve effectively. It is not known which Zariski surfaces are liftable from characteristic p to characteristic zero in the sense of Grothendieck and Lubkin and of course the same is true for Zariski B surfaces. A result of Deligne implies that K3 type Zariski surfaces are liftable. Results of Brieskorn and Michael Artin give some local information about liftability of generic Zariski surfaces that arise from resolving rational singularities.

Computer mathematics

Computer algebra has been used extensively to compute Picard groups of Zariski surfaces. After seminal work of Jacobson,Cartier,Samuel and Jeffey Lang in his Purdue Ph.D. thesis 1980 the computer program was created by David Joyce in Pascal. Students of Jeffrey Lang at the University of Kansas have simplified this program and expressed it using Wolfram Mathematica language and system.

Another possible interface with computer mathematics is creating data bases and even data warehouses containing Zariski surfaces defined over finite fields. Technology would be developed to compute the usual genera and plurigenera of the surfaces stored in the data base or data warehouse as well as a number of other numerical invariants for example the Artin invariant could be computed on stored. This could lead to a visualization of the moduli spaces including the stratification according to the Artin invariant.Michael Artin conjectures deep interpretation of this picture as a kind of period map with the geometric genus of the surface playing a major role in the dimensions of the strata.

Recent work

Torsten Ekedahl from Sweden computed the crystalline cohomology of Zariski surfaces in some cases. Ofer Gaber and Ray Hoobler studied with Piotr Blass the Brauer group of Zariski surfaces This work is considered inconclusive as of today 2006.

Oscar Zariski and Piotr Blass revived the theory of adjoint surfaces created by the Italian geometers to compute numerical invariants of Zariski surfaces. This has been continued by Joseph Lipmann and also more recently by the computer algebra group in Linz Austria mainly by Joseph Schicho. Currently attempts are being made to use Zariski surfaces for coding ,encryption and also for mathematical physics applications. As of 2006 these efforts are inconclusive.

Open problems

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? 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 Ph.D. thesis and by William E. Lang in his Harvard Ph.D. thesis in 1978.

As mentioned above Zariski surfaces are birational to surfaces in affine 3-space A defined by irreducible polynomials of the form

z = f(x, y).

There is ample evidence to conjecture that for p>=5 and for a general choice of the polynomial f(x,y) the above affine surface is factorial. Thus Zariski surfaces conjecturally give rise to a large family of two dimensional factorial rings. It would be intersesting to apply these rings to data encryption theory and practice.

Zariski threefolds and manifolds of higher dimension have been similarly defined and a broad theory is slowly emerging as of 2006 in preprint form. There is only a very rudimentary theory of moduli of Zariski surfaces to be further developed as of 2006.

See also

• List of algebraic surfaces

References

• Zariski Surfaces And Differential Equations in Characteristic p > 0 by Piotr Blass, Jeffrey Lang ISBN 0-8247-7637-2
• Blass, Piotr; Lang, Jeffrey Surfaces de Zariski factorielles. (Factorial Zariski surfaces). C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), no. 15, 671--674.
• Zariski, Oscar On Castelnuovo's criterion of rationality p_a=P₂=0 of an algebraic surface. Illinois J. Math. 2 1958 303--315.

• Algebraic varieties
• Algebraic surfaces