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 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. (Wikipedia).
Ananth Shankar, Picard ranks of K3 surfaces and the Hecke orbit conjecture
VaNTAGe Seminar, November 23, 2021
From playlist Complex multiplication and reduction of curves and abelian varieties
J. Bost - Techniques d’algébrisation... (Part 2)
Abstract - Dans ce cours, nous nous proposons d’expliquer comment des théorèmes d’algébrisation classiques, concernant des variétés ou des faisceux cohérents analytiques, possèdent des avatars en géométrie formelle et en géométrie diophantienne. Nous mettrons l’accent sur les points commun
From playlist Ecole d'été 2019 - Foliations and algebraic geometry
Geometry of complex surface singularities and 3-manifolds - Neumann
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From playlist Mathematics
E. Amerik - On the characteristic foliation
Abstract - Let X be a holomorphic symplectic manifold and D a smooth hypersurface in X. Then the restriction of the symplectic form on D has one-dimensional kernel at each point. This distribution is called the characteristic foliation. I shall survey a few results concerning the possible
From playlist Ecole d'été 2019 - Foliations and algebraic geometry
algebraic geometry 5 Affine space and the Zariski topology
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers the definition of affine space and its Zariski topology.
From playlist Algebraic geometry I: Varieties
Integral points on Markoff-type cubic surfaces - Amit Ghosh
Special Seminar Topic: Integral points on Markoff-type cubic surfaces Speaker: Amit Ghosh Affiliation: Oklahoma State University Date: December 8, 2017 For more videos, please visit http://video.ias.edu
From playlist Mathematics
Commensurators of thin Subgroups by Mahan M. J.
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From playlist Smooth And Homogeneous Dynamics
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This talk is part of a series about complex surfaces, and explains what minimal surfaces are. A minimial surfaces is one that cannot be obtained by blowing up a nonsingular surfaces at a point. We explain why every surface is birational to a minimal nonsingular projective surface. We disc
From playlist Algebraic geometry: extra topics
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From playlist English interviews - Interviews en anglais
Marta Pieropan, The split torsor method for Manin’s conjecture
See https://tinyurl.com/y98dn349 for an updated version of the slides with minor corrections. VaNTAGe seminar 20 April 2021
From playlist Manin conjectures and rational points