On a question of Zariski on Zariski surfaces

We give a new method to construct unirational surfaces which may be applied to the following question posed by Zariski in his studies on unirational surfaces. Is any Zariski surface with geometric genus zero rational? Our main result is a negative answer to this question in any characteristic case.     

Zariski geometries

A characterization is obtained of the Zariski topology over an algebraically closed field.

     

On a question ofMakar-Limanov

Let be an uncountable field, let be a field extension, and let be an associative -algebra. We show that if contains a non-commutative free algebra, then so does .

     

On existence of log minimal models and weak Zariski decompositions

We first introduce a weak type of Zariski decomposition in higher dimensions: an -Cartier divisor has a weak Zariski decomposition if birationally and in a numerical sense it can be written as the sum of a nef and an effective -Cartier divisor. We then prove that there is a very basic relation between Zariski decompositions and log minimal models which has long been expected: we prove that assuming the log minimal model program in dimension d − 1, a lc pair (X/Z, B) of dimension d has a log minimal model (in our sense) if and only if K _X  + B has a weak Zariski decomposition/Z.     

On a question of Gross

Using the notion of weighted sharing of sets we prove two uniqueness theorems which improve the results proved by Fang and Qiu [H. Qiu, M. Fang, A unicity theorem for meromorphic functions, Bull. Malaysian Math. Sci. Soc. 25 (2002) 31-38], Lahiri and Banerjee [I. Lahiri, A. Banerjee, Uniqueness of meromorphic functions with deficient poles, Kyungpook Math. J. 44 (2004) 575-584] and Yi and Lin [H.X. Yi, W.C. Lin, Uniqueness theorems concerning a question of Gross, Proc. Japan Acad. Ser. A 80 (2004) 136-140] and thus provide an answer to the question of Gross [F. Gross, Factorization of meromorphic functions and some open problems, in: Proc. Conf. Univ. Kentucky, Lexington, KY, 1976, in: Lecture Notes in Math., vol. 599, Springer, Berlin, 1977, pp. 51-69], under a weaker hypothesis.     

On a question of Zhang

When studying the conjugacy class version of the Huppert’s ρ-σ conjecture, Jiping Zhang raised a number theory question. In this article, we provide examples to show that the question raised by Zhang is not always true in general.     

On a question of Goss

In this note we answer the question raised by D. Goss in [D. Goss, Applications of non-Archimedean integration to the L-series of τ-sheaves, J. Number Theory 110 (1) (2005) 83-113] by proving that the group of locally analytic endomorphisms on the 1-units of a locally compact field of characteristic p>0 is isomorphic to the p-adic integers.     

On a question of Pereira

Answering a question of Pereira we show that it is possible to have a model violating the Singular Cardinal Hypothesis without a tree-like continuous scale. We are grateful to Luis Pereira for asking an interesting question and his comments on the paper. We would like to thank to the referee of the paper for his comments and corrections.     

On a question of chambers and makan

This work has been supported by Proyecto PB 90-0414-C03-01 of DGICYT, Ministerio de Educación Ciencia of Spain.     

On a question of Slaman and Groszek

We answer a question of Slaman and Groszek by showing that any non-computable perfect tree computes one of its non-computable paths.

     

On a question of Brezis and Marcus

Motivated by a question of Brezis and Marcus, we show that the L^p–Hardy inequality involving the distance to the boundary of a convex domain, can be improved by adding an L^q norm q ≥ p, with a constant depending on the interior diameter of Ω.