The place of exceptional covers among all diophantine relations

A cover of normal varieties is exceptional over a finite field if the map on points over infinitely many extensions of the field is one-one. A cover over a number field is exceptional if it is exceptional over infinitely many residue class fields. The first result: The category of exceptional covers of a normal variety, Z, over a finite field, F_q, has fiber products, and therefore a natural Galois group (with permutation representation) limit. This has many applications to considering Poincare series attached to diophantine questions. The paper follows three lines: * The historical role of the Galois Theoretic property of exceptionality, first considered by Davenport and Lewis. * How the tower structure on the category of exceptional covers of a pair (Z,F_q) allows forming subtowers that separate known results from unknown territory. * The use of Serre's OIT, especially the GL_2 case, to consider cryptology periods and functional composition aspects of exceptionality. A more extensive html description of the paper is at http://www.math.uci.edu/~mfried/paplist-ff/exceptTowYFFTA_519.htm

A survey of Elekes-R\'onyai-type problems

We give an overview of recent progress around a problem introduced by Elekes and R\'onyai. The prototype problem is to show that a polynomial $f\in \mathbb{R}[x,y]$ has a large image on a Cartesian product $A\times B\subset \mathbb{R}^2$, unless $f$ has a group-related special form. We discuss a number of variants and generalizations. This includes the Elekes-Szab\'o problem, which generalizes the Elekes-R\'onyai problem to a question about an upper bound on the intersection of an algebraic surface with a Cartesian product, and curve variants, where we ask the same questions for Cartesian products of finite subsets of algebraic curves. These problems lie at the crossroads of combinatorics, algebra, and geometry: They ask combinatorial questions about algebraic objects, whose answers turn out to have applications to geometric questions involving basic objects like distances, lines, and circles, as well as to sum-product-type questions from additive combinatorics. As part of a recent surge of algebraic techniques in combinatorial geometry, a number of quantitative and qualitative steps have been made within this framework. Nevertheless, many tantalizing open questions remain

A numerical toolkit for multiprojective varieties

A numerical description of an algebraic subvariety of projective space is given by a general linear section, called a witness set. For a subvariety of a product of projective spaces (a multiprojective variety), the corresponding numerical description is given by a witness collection, whose structure is more involved. We build on recent work to develop a toolkit for the numerical manipulation of multiprojective varieties that operates on witness collections, and use this toolkit in an algorithm for numerical irreducible decomposition of multiprojective varieties. The toolkit and decomposition algorithm are illustrated throughout in a series of examples.Comment: 28 page

Invariants of D(q,p) singularities

The basic examples of functions defining non-isolated hypersurface singularities are the A(d) singularities and the D(q,p) singularities. The A(d) singularities, up to analytic equivalence, are the product of a Morse function and the zero map, while the simplest D(q,p) singularity is the Whitney umbrella. These are the basic examples, because they correspond to stable germs of functions in the study of germs of functions with non-isolated singularities. Given a germ of a function which defines a non-isolated hypersurface singularity at the origin, which in the appropriate sense, has finite codimension in the set of such germs, the singularity type of such germs away from the origin is A(d) or D(q,p). In this note we calculate the homotopy type of the Milnor fiber of germs of type D(q,p), as well as their L\^e numbers. The calculation of the L\^e numbers involves the use of an incidence variety which may be useful for studying germs of finite codimension. The calculation shows that the set of symmetric matrices of kernel rank greater than or equal to 1 is an example of a hypersurface singularity with a Whitney stratification (given by the rank of the matrices) in which only one singular stratum gives a component of top dimension of the singular set of the conormal.Comment: 10 pages, corrects typos and a small mistake in the proof of theorem 1.1. Several proofs are expanded to improve readability. Material on the Euler obstruction adde

Idempotents in representation rings of quivers

For an acyclic quiver Q, we solve the Clebsch-Gordan problem for the projective representations by computing the multiplicity of a given indecomposable projective in the tensor product of two indecomposable projectives. Motivated by this problem for arbitrary representations, we study idempotents in the representation ring of Q (the free abelian group on the indecomposable representations, with multiplication given by tensor product). We give a general technique for constructing such idempotents and for decomposing the representation ring into a direct product of ideals, utilizing morphisms between quivers and categorical Moebius inversion.Comment: 30 pages, v2: small improvements in expositio

Seshadri constants via Lelong numbers

One of Demailly's characterizations of Seshadri constants on ample line bundles works with Lelong numbers of certain positive singular hermitian metrics. In this note sections of multiples of the line bundle are used to produce such metrics and then to deduce another formula for Seshadri constants. It is applied to compute Seshadri constants on blown up products of curves, to disprove a conjectured characterization of maximal rationally connected quotients and to introduce a new approach to Nagata's conjecture.Comment: 12 pages; improved exposition and new results related to Nagata's Conjectur

Moduli of weighted stable maps and their gravitational descendants

We study the intersection theory on the moduli spaces of maps of $n$-pointed curves $f:(C,s_1,... s_n)\to V$ which are stable with respect to a weight data $(a_1,..., a_n)$, $0\le a_i\le 1$. After describing the structure of these moduli spaces, we prove a formula describing the way each descendant changes under a wall crossing. As a corollary, we compute the weighted descendants in terms of the usual ones, i.e. for the weight data $(1,...,1)$, and vice versa.Comment: v3: mostly typographical edits; v4: minor update following referee's comment

Inequivalent Lefschetz fibrations and surgery equivalence of symplectic 4-manifolds

We prove that any symplectic 4-manifold which is not a rational or ruled surface, after sufficiently many blow-ups, admits an arbitrary number of nonisomorphic Lefschetz fibrations of the same genus which cannot be obtained from one another via Luttinger surgeries. This generalizes results of Park and Yun who constructed pairs of nonisomorphic Lefschetz fibrations on knot surgered elliptic surfaces. In turn, we prove that there are monodromy factorizations of Lefschetz pencils which have the same characteristic numbers but cannot be obtained from each other via partial conjugations by Dehn twists, answering a problem posed by Auorux.Comment: 13 pages, a couple more corrections and clarifications for publicatio

Spaces of Incompressible Surfaces

This is a "software upgrade" to a paper originally published in 1976, with cleaner statements and improved proofs. The main result is that, in a Haken 3-manifold, the space of all incompressible surfaces in a single isotopy class is contractible, except when the surface is the fiber of a surface bundle structure, in which case the space of all surfaces isotopic to the fiber has the homotopy type of a circle (the fibers). The main application from the 1976 paper is also rederived, the theorem (proved independently by Ivanov) that the diffeomorphism group of a Haken 3-manifold has contractible components, except in the case of certain Seifert manifolds when the components of the diffeomorphism group have the homotopy type of a circle or torus acting on the manifold.Comment: 10 page

Partially hyperbolic diffeomorphisms with compact center foliations

Let f:M->M be a partially hyperbolic diffeomorphism such that all of its center leaves are compact. We prove that Sullivan's example of a circle foliation that has arbitrary long leaves cannot be the center foliation of f. This is proved by thorough study of the accessible boundaries of the center-stable and the center-unstable leaves. Also we show that a finite cover of f fibers over an Anosov toral automorphism if one of the following conditions is met: 1. the center foliation of f has codimension 2, or 2. the center leaves of f are simply connected leaves and the unstable foliation of f is one-dimensional.Comment: 22 pages, 1 figure. In the second version an error was corrected, the exposition was improved, new references adde