Zariski surface

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 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"³ defined by irreducible polynomials of the form

:"z^p" = "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 a rational surface, and 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, now called the Artin invariant, that gives a 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 the David Mumford school, the Picard scheme need not be reduced (reference: 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 required modification of the classical theorem of Bertini in characteristic "p" > 0. Research about Zariski surfaces led to exploration of theorems 'of Bertini type'. Grothendieck allowed his notes on Bertini type theorems to be published under the title EGA 5; they are vailable translated and partially edited t the website of the Grothendieck circle centered in France and at the website of James Milne of 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", a Zariski B surface means any smooth projective surface S whose function fieldarises from the function field of B by adjoining a p-th root. When B is an abelian surface, Barsotti and Mumford developed examples and a partialtheory prior to 1980. Another case is when B is a surface of general type with infinite cyclic Picard group 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 Saul Lubkin.

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 Jeffrey Lang in his Purdue Ph.D. thesis 1980, a computer program was created by David Joyce, in Pascal. Students of Jeffrey Lang at the University of Kansas have simplified this program and expressedit using Wolfram Mathematica language and system.

Further progress could lead to a visualization of the moduli spaces including the stratification according to the Artin invariant. Michael Artin conjectures an 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.

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"³ defined by irreducible polynomials of the form

:"z^p" = "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 should give rise to a large family of two dimensional
factorial rings.

Zariski threefolds and manifolds of higher dimension have been similarly defined and atheory is slowly emerging.

ee 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.