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 in 1977 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
The following problem was posed by Oscar Zariski in 1971: 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. Kentaro Mitsui (2014) 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.
In algebraic geometry, a branch of mathematics, a Zariskisurface is a surface over a field of characteristic p > 0 such that there is a dominant inseparable...
Zariski ring Zariski tangent space ZariskisurfaceZariski topology Zariski–Riemann surfaceZariski space (disambiguation) Zariski's lemma Zariski's main...
used in the Enriques–Kodaira classification to identify the rational surfaces. Zariski (1958) proved that Castelnuovo's theorem also holds over fields of...
Further such examples arise in Zariskisurface theory. He also conjectures that the Kodaira vanishing theorem is false for surfaces in characteristic p. In the...
Enriques surface K3 surface Hodge index theorem Elliptic surfaceSurface of general type Zariskisurface Algebraic variety Hypersurface Quadric (algebraic geometry)...
{F}}_{g}} of polarized complex K3 surfaces of genus g for each g ≥ 2 {\displaystyle g\geq 2} ; it can be viewed as a Zariski open subset of a Shimura variety...
V. (2001) [1994], "Algebraic surface", Encyclopedia of Mathematics, EMS Press Zariski, Oscar (1995), Algebraic surfaces, Classics in Mathematics, Berlin...
More precisely, the images of open sets in the Zariski topology are again open. The Veronese surface is the only Severi variety of dimension 2 Joe Harris...
both the arithmetic genus and the second plurigenus. Zariski found some examples (Zariskisurfaces) in characteristic p > 0 that are unirational but not...
mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples...
resolution for surfaces over the complex numbers by Del Pezzo (1892), Levi (1899), Severi (1914), Chisini (1921), and Albanese (1924), but Zariski (1935, chapter...
meaning, applying to projective curves and compact Riemann surfaces in particular. A Zariski geometry consists of a set X and a topological structure on...
Research played a significant role in computing the Picard group of a Zariskisurface via the work of Jeffrey Lang and collaborators. The method was inspired...
Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of P n {\displaystyle \mathbb {P} ^{n}} . A projective...
include mod p phenomena by Zariski, the Shafarevich school and others by around 1960. The form of the Riemann–Roch theorem on a surface was also worked out....
other words that the Zariski–Riemann space of the array is in some sense non-singular. Local uniformization was introduced by Zariski (1939, 1940), who separated...
fundamental group is used as a replacement for the fundamental group. Since the Zariski topology on an algebraic variety or scheme X is much coarser than, say...
considered is the usual one in the case of a real or complex quadric, or the Zariski topology in all cases). The points of the quadric that are not in the image...
key result is that the Schubert cells (or rather, the classes of their Zariski closures, the Schubert cycles or Schubert varieties) span the whole cohomology...
thesis, written under the direction of Oscar Zariski, was titled Local uniformization on algebraic surfaces over modular ground fields. Before going to...
valuations of K/k. These sets can be given a natural topological structure: the Zariski–Riemann space of K/k. function field of an algebraic variety function field...
in general; the Zariski topology on it will have more closed sets (except in very simple cases). See also Segre embedding. Zariski, Oscar (1958). "Introduction...
simplex and every simplicial complex inherits a natural topology from . The Zariski topology is defined algebraically on the spectrum of a ring or an algebraic...
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