Automorphisms of smooth canonically polarized surfaces in positive characteristic

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Number Theory, Analysis and Geometry: In Memory of Serge Lang

Serge Lang was an iconic figure in mathematics, both for his own important work and for the indelible impact he left on the field of mathematics, on his students, and on his colleagues. Over the course of his career, Lang traversed a tremendous amount of mathematical ground. As he moved from subject to subject, he found analogies that led to important questions in such areas as number theory, arithmetic geometry and the theory of negatively curved spaces. Lang's conjectures will keep many mathematicians occupied far into the future. In the spirit of Lang’s vast contribution to mathematics, this memorial volume contains articles by prominent mathematicians in a variety of areas, namely number theory, analysis and geometry, representing Lang’s own breadth of interests. A special introduction by John Tate includes a brief and engaging account of Serge Lang’s life

Progress in Commutative Algebra 2

This is the second of two volumes of a state-of-the-art survey article collection which originates from three commutative algebra sessions at the 2009 Fall Southeastern American Mathematical Society Meeting at Florida Atlantic University. The articles reach into diverse areas of commutative algebra and build a bridge between Noetherian and non-Noetherian commutative algebra. These volumes present current trends in two of the most active areas of commutative algebra: non-noetherian rings (factorization, ideal theory, integrality), and noetherian rings (the local theory, graded situation, and interactions with combinatorics and geometry). This volume contains surveys on aspects of closure operations, finiteness conditions and factorization. Closure operations on ideals and modules are a bridge between noetherian and nonnoetherian commutative algebra. It contains a nice guide to closure operations by Epstein, but also contains an article on test ideals by Schwede and Tucker and more

An Explicit Construction of Sheaves in Context

This document details the body of theory necessary to explicitly construct sheaves of sets on a site together with the development of supporting material necessary to connect sheaf theory with the wider mathematical contexts in which it is applied. Of particular interest is a novel presentation of the plus construction suitable for direct application to a site without first passing to the generated grothendieck topology

Number Theory, Analysis and Geometry: In Memory of Serge Lang

Serge Lang was an iconic figure in mathematics, both for his own important work and for the indelible impact he left on the field of mathematics, on his students, and on his colleagues. Over the course of his career, Lang traversed a tremendous amount of mathematical ground. As he moved from subject to subject, he found analogies that led to important questions in such areas as number theory, arithmetic geometry and the theory of negatively curved spaces. Lang's conjectures will keep many mathematicians occupied far into the future. In the spirit of Lang’s vast contribution to mathematics, this memorial volume contains articles by prominent mathematicians in a variety of areas, namely number theory, analysis and geometry, representing Lang’s own breadth of interests. A special introduction by John Tate includes a brief and engaging account of Serge Lang’s life

Approximation Constants for Closed Subschemes of Projective Varieties

Diophantine approximation is a branch of number theory with a long history, going back at least to the work of Dirichlet and Liouville in the 1840s. The innocent-looking question of how well an arbitrary real algebraic number can be approximated by rational numbers (relative to the size of the denominator of the approximating rational number) took more than 100 years to resolve, culminating in the definitive Fields Medal-winning work of Klaus Roth in 1955. Much more recently, David McKinnon and Mike Roth have re-phrased and generalized this Diophantine approximation question to apply in the setting of approximating algebraic points on projective varieties defined over number fields. To do this, they defined an "approximation constant", depending on the point one wishes to approximate and a given line bundle. This constant measures the tradeoff between the closeness of the approximation and the arithmetic complexity of the point used to make the approximation, as measured by a height function associated to the line bundle. In particular, McKinnon and Roth succeeded in proving lower bounds on the approximation constant in terms of the "Seshadri constant" associated to the given point and line bundle, measuring local positivity of the line bundle around the point. Appropriately interpreted, these results generalize the classical work of Liouville and Roth, and the corresponding McKinnon-Roth theorems are therefore labelled "Liouville-type" and "Roth-type" results. Recent work of Grieve and of Ru-Wang have taken the Roth-type theorems even further; in contrast, we explore results of Liouville-type, which are more elementary in nature. In Chapter 2, we lay the groundwork necessary to define the approximation constant at a point, before generalizing the McKinnon-Roth definition to approximations of arbitrary closed subschemes. We also introduce the notion of an essential approximation constant, which ignores unusually good approximations along proper Zariski-closed subsets. After verifying that our new approximation constant truly does generalize the constant of McKinnon-Roth, Chapter 3 establishes a fundamental lower bound on the approximation constants of closed subschemes of projective space, depending only on the equations cutting out the subscheme. In Chapter 4, we provide a series of explicit computations of approximation constants, both for subschemes satisfying suitable geometric conditions, and for curves of low degree in projective 3-space. We will encounter difficulties computing the approximation constant exactly for general cubic curves, and we spend some time showing why some of the more evident approaches do not succeed. To conclude the chapter, we take up the question of large gaps between the ordinary and essential approximation constants, by considering approximations to a certain rational point on a diagonal quartic surface. Finally, in Chapter 5, we generalize the Liouville-type results of McKinnon-Roth