Zariski surface

For spaces of valuations, see Zariski–Riemann 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 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 polynomials of the form

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

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?

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