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We investigate the role played by curve singularity germs in the enumeration of inflection points in families of curves acquiring singular members. Let

We define a class of random measures, spatially independent martingales, which we view as a natural generalization of the canonical random discrete set, and which includes as special cases many variants of fractal percolation and Poissonian cut-outs. We pair the random measures with deterministic families of parametrized measures

This book considers the so-called Unlikely Intersections, a topic that embraces well-known issues, such as Lang's and Manin-Mumford's, concerning torsion points in subvarieties of tori or abelian varieties. More generally, the book considers algebraic subgroups that meet a given subvariety in a set ofunlikelydimension. The book is an expansion of the Hermann Weyl Lectures delivered by Umberto Zannier at the Institute for Advanced Study in Princeton in May 2010. The book consists of four chapters and seven brief appendixes, the last six by David Masser. The first chapter considers multiplicative algebraic groups, presenting proofs of several developments, ranging from the origins to recent results, and discussing many applications and relations with other contexts. The second chapter considers an analogue in arithmetic and several applications of this. The third chapter introduces a new method for approaching some of these questions, and presents a detailed application of this (by Masser and the author) to a relative case of the Manin-Mumford issue. The fourth chapter focuses on the André-Oort conjecture (outlining work by Pila).

A family {Dq} of complexes of free R-modules is constructed for each almost alternating matrix p = (XY) of shape (g,f) over a commutative noetherian ring R. (In p,X is a g x g alternating matrix and Y is an arbitrary g x f matrix). The complexes Dq are in many ways analogous to the Buchsbaum-Rim and Eagon-Northcott complexes which are associated to a generic matrix. For example, the complex Dq is isomorphic to the (shifted) dual of the complex Df-2-q; and the complex Dq can be obtained as the mapping cone of two of the complexes which correspond to an almost alternating matrix of shape (g, f - 1). Roughly speaking, the complex Dq is obtained by pasting a graded strand of algebra K/J (where K is the Koszul algebra associated to p and J is a two generated ideal of K, together with a different graded strand of the same algebra. The position in Dq where the two strands are patched together involves pfaffians, of various sizes, of the alternating map which corresponds to p. If p is sufficiently general, then Dq is acyclic for all g 5 - 1.;If the maximal order pfaffians of X generate a grade three Gorenstein ideal I of R and f 5 3, then Do resolves R/J, where J is an f-residual intersection of I. In the generic case, the divisor class group of R/J is the infinite cyclic group generated by the cokernel of p, and Dq resolves a representative of the class q(cockerp) from C M(R/J) for all q 5-1. When f = O, then Dq resolves the qth poser Iq of the grade three Gorenstein ideal I.

1989 marked the 150th anniversary of the birth of the great Danish mathematician Hieronymus George Zeuthen. Zeuthen's name is known to every algebraic geometer because of his discovery of a basic invariant of surfaces. However, he also did fundamental research in intersection theory, enumerative geometry, and the projective geometry of curves and surfaces. Zeuthen's extraordinary devotion to his subject, his characteristic depth, thoroughness, and clarity of thought, and his precise and succinct writing style are truly inspiring. During the past ten years or so, algebraic geometers have reexamined Zeuthen's work, drawing from it inspiration and new directions for development in the field. The 1989 Zeuthen Symposium, held in the summer of 1989 at the Mathematical Institute of the University of Copenhagen, provided a historic opportunity for mathematicians to gather and examine those areas in contemporary mathematical research which have evolved from Zeuthen's fruitful ideas. This volume, containing papers presented during the symposium, as well as others inspired by it, illuminates some currently active areas of research in enumerative algebraic geometry.

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From the ancient origins of algebraic geometry in the solutions of polynomial equations, through the triumphs of algebraic geometry during the last two centuries, intersection theory has played a central role.The aim of this book is to develop the foundations of this theory, and to indicate the range of classical and modern applications.