Zariski surface

In algebraic geometry, a branch of mathematics, a Zariski surface, named after Oscar Zariski who found some examples in 1958, 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. However they are not all rational, and in particular they give examples of unirational surfaces in characteristic p>0 that are not rational. (In characteristic 0, Castelnuovo's theorem implies that all unirational surfaces are rational.)

Zariski surfaces are birational to surfaces in affine 3-space A³ defined by irreducible polynomials of the form

z^p = f(x, y).

Piotr Blass and Jeff Lang have computed the Picard group of the generic Zariski surface using some ideas of Pierre Deligne and Alexander Grothendieck during 1980-1993.

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?

It should be noted that 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. It has been shown that 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, Kummer-Kodaira-Kahler or K3 surfaces,Enriques surfaces ,quasi elliptic surfaces and also rational surfaces. In every characteristic the family of birationally distinct Zariski surfaces is infinite. 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.

An extensive bibliography of books and papers on Zariski surfaces can be found at [[1]]

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.