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• Rational Equivalence.- Divisors.- Vector Bundles.- Cones and Segre Classes.- Deformations to the Normal Cone.- Intersection Products.- Intersection Multiplicites.- Intersections on Non-singular Varieties.- Excess and Residual Intersections.- Families of Algebraic Cycles.- Dynamic Intersections.- Positivity.- Rationality.- Degeneracy Loci and Grassmannians.- Riemann-Roch for Non-singular Varieties.- Correspondences.- Bivariant Intersections Theory.- Riemann-Roch for Singular Varieties.- Algebraic: Homological and Numerical Equivalence.- Generalizations.
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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. Although a comprehensive history of this vast subject is not attempted, the author points out some of the striking early appearances of the ideas of intersection theory. A suggested prerequisite for the reading of this book is a first course in algebraic geometry. Fulton's introduction to intersection theory has been well used for more than 10 years. It is still the only existing complete modern treatise of the subject and received the Steele Prize for best exposition in August 1996.
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The main themes of this book are to establish the triple formula without any hypotheses on the genericity of the morphism, and to develop a theory of complete quadruple points, which is a first step towards proving the quadruple point formula under less restrictive hypotheses.This book should be of interest to graduate students and researchers in the field of algebraic geometry. The reader is expected to have some basic knowledge of enumerative algebraic geometry and pointwise Hilbert schemes.
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• Intersections of hypersurfaces Multiplicity and normal cones Divisors and rational equivalence Chern classes and Segre classes Gysin maps and intersection rings Degeneracy Loci Refinements Positivity Riemann-Roch Miscellany References Notes (1983-1995).
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This book introduces some of the main ideas of modern intersection theory, traces their origins in classical geometry and sketches a few typical applications. It requires little technical background: much of the material is accessible to graduate students in mathematics. A broad survey, the book touches on many topics, most importantly introducing a powerful new approach developed by the author and R. MacPherson. It was written from the expository lectures delivered at the NSF-supported CBMS conference at George Mason University, held June 27-July 1, 1983.The author describes the construction and computation of intersection products by means of the geometry of normal cones. In the case of properly intersecting varieties, this yields Samuel's intersection multiplicity; at the other extreme it gives the self-intersection formula in terms of a Chern class of the normal bundle; in general it produces the excess intersection formula of the author and R. MacPherson. Among the applications presented are formulas for degeneracy loci, residual intersections, and multiple point loci; dynamic interpretations of intersection products; Schubert calculus and solutions to enumerative geometry problems; and Riemann-Roch theorems.
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• Preliminaries.- Patching Modules and Other Preliminaries.- Extended Modules over Polynomial Rings.- Modules over Commutative Rings.- The Theory of Matrices.- Complete Intersections.- The Techniques of Lindel.- Bibliography.- Index.
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In these notes on "Projective Modules and Complete Intersections" an account on the recent developments in research on this subject is presented. The author's preference for the technique of Patching isotopic isomorphisms due to Quillen, formalized by Plumsted, over the techniques of elementary matrices is evident here. The treatment of Basic Element theory here incorporates Plumstead's idea of the "generalized dimension functions". These notes are highly selfcontained and should be accessible to any graduate student in commutative algebra or algebraic geometry. They include fully self-contained presentations of the theorems of Ferrand-Szpiro, Cowsik-Nori and the techniques of Lindel.
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• *FrontMatter, pg. i*Contents, pg. v*Preface, pg. ix*Notation and Conventions, pg. xi*Introduction: An Overview of Some Problems of Unlikely Intersections, pg. 1*
• Chapter 1: Unlikely Intersections in Multiplicative Groups and the Zilber Conjecture, pg. 15*
• Chapter 2: An Arithmetical Analogue, pg. 43*
• Chapter 3 Unlikely Intersections in Elliptic Surfaces and Problems of Masser, pg. 62*
• Chapter 4: About the Andre-Oort Conjecture, pg. 96*Appendix A: Distribution of Rational Points on Subanalytic Surfaces, pg. 128*Appendix B: Uniformity in Unlikely Intersections: An Example for Lines in Three Dimensions, pg. 136*Appendix C: Silverman's Bounded Height Theorem for Elliptic Curves: A Direct Proof, pg. 138*Appendix D: Lower Bounds for Degrees of Torsion Points: The Transcendence Approach, pg. 140*Appendix E: A Transcendence Measure for a Quotient of Periods, pg. 143*Appendix F: Counting Rational Points on Analytic Curves: A Transcendence Approach, pg. 145*Appendix G: Mixed Problems: Another Approach, pg. 147*Bibliography, pg. 149*Index, pg. 159.
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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 of unlikely dimension. 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 Andre-Oort conjecture (outlining work by Pila).
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• The Classical Bezout Theorem.- The Intersection Algorithm and Applications.- Connectedness and Bertini Theorems.- Joins and Intersections.- Converse to Bezout's Theorem.- Intersection Numbers and their Properties.- Linkage, Koszul Cohomology and Intersections. - Further Applications.- Appendix.
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The central topic of the book is refined Intersection Theory and its applications, the central tool of investigation being the Stuckrad-Vogel Intersection Algorithm, based on the join construction. This algorithm is used to present a general version of Bezout's Theorem, in classical and refined form. Connections with the Intersection Theory of Fulton-MacPherson are treated, using work of van Gastel employing Segre classes. Bertini theorems and Connectedness theorems form another major theme, as do various measures of multiplicity. We mix local algebraic techniques as e.g. the theory of residual intersections with more geometrical methods, and present a wide range of geometrical and algebraic applications and illustrative examples. The book incorporates methods from Commutative Algebra and Algebraic Geometry and therefore it will deepen the understanding of Algebraists in geometrical methods and widen the interest of Geometers in major tools from Commutative Algebra.
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• Introduction to degeneracy loci and Schubert polynomials-- modern formulation - Grassmannians, flag varieties, Schubert varieties-- symmetric polynomials useful in geometry-- polynomials supported on degeneracy loci-- the Euler characteristic of degeneracy loci-- flag manifolds and determinantal formulas for other classical groups-- P- and Q-polynomial formulas for other classical groups-- the classes of Brill-Noether loci in Prym varieties-- applications and open problems.
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Schubert varieties and degeneracy loci have a long history in mathematics, starting from questions about loci of matrices with given ranks. These notes, taken from a summer school in Thurnau, aim to give an introduction to these topics, and to describe late-1990s progress on these problems. There are interesting interactions with the algebra of symmetric functions and combinatorics, as well as the geometry of flag manifolds and intersections theory and algebraic geometry.
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• Introduction The universal pseudo-quotient for a family of subvarieties Normal bundles of quadrics in $X$ Morphisms from quadrics to Grassmannians Pointwise uniform vector bundles on non-singular quadrics Theory of extensions of families over Hilbert schemes Existence of algebraic quotient--proof of Theorem 0.3 Appendix. Deformations of vector bundles on infinitesimally rigid projective varieties with null global $i$-forms References.
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• *FrontMatter, pg. i*Contents, pg. v*Preface, pg. ix*Notation and Conventions, pg. xi*Introduction: An Overview of Some Problems of Unlikely Intersections, pg. 1*
• Chapter 1: Unlikely Intersections in Multiplicative Groups and the Zilber Conjecture, pg. 15*
• Chapter 2: An Arithmetical Analogue, pg. 43*
• Chapter 3 Unlikely Intersections in Elliptic Surfaces and Problems of Masser, pg. 62*
• Chapter 4: About the Andre-Oort Conjecture, pg. 96*Appendix A: Distribution of Rational Points on Subanalytic Surfaces, pg. 128*Appendix B: Uniformity in Unlikely Intersections: An Example for Lines in Three Dimensions, pg. 136*Appendix C: Silverman's Bounded Height Theorem for Elliptic Curves: A Direct Proof, pg. 138*Appendix D: Lower Bounds for Degrees of Torsion Points: The Transcendence Approach, pg. 140*Appendix E: A Transcendence Measure for a Quotient of Periods, pg. 143*Appendix F: Counting Rational Points on Analytic Curves: A Transcendence Approach, pg. 145*Appendix G: Mixed Problems: Another Approach, pg. 147*Bibliography, pg. 149*Index, pg. 159.
• (source: Nielsen Book Data)

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 of unlikely dimension. 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 Andre-Oort conjecture (outlining work by Pila).
(source: Nielsen Book Data)

• Coxeter Groups and the Davis Complex (T.A. Schroeder).- A Tale of Two Symmetries: Embeddable and Non-Embeddable Group Actions on Surfaces (V. Peterson, A. Wootton).- Tile Invariants for Tackling Tiling Questions (M.P. Hitchman).- Forbidden Minors: Finding the Finite Few (T.W. Mattman).- Introduction to competitive graph coloring (C. Dunn, V. Larsen, J.F. Nordstrom).- Matrioids (E. McNicholas, N.A. Neudauer, C. Starr).- Finite Frame Theory (S. Datta, J. Oldroyd).- Mathematical decision-making with linear and convex programming (J. Kotas).- Computing weight multiplicities (P. E. Harris).- Vaccination strategies for small worlds. (W. Just, H. C. Highlander).- Steady and Stable: Numerical Investigations of Nonlinear Partial Differential Equations (R. C. Harwood).
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This highly readable book aims to ease the many challenges of starting undergraduate research. It accomplishes this by presenting a diverse series of self-contained, accessible articles which include specific open problems and prepare the reader to tackle them with ample background material and references. Each article also contains a carefully selected bibliography for further reading. The content spans the breadth of mathematics, including many topics that are not normally addressed by the undergraduate curriculum (such as matroid theory, mathematical biology, and operations research), yet have few enough prerequisites that the interested student can start exploring them under the guidance of a faculty member. Whether trying to start an undergraduate thesis, embarking on a summer REU, or preparing for graduate school, this book is appropriate for a variety of students and the faculty who guide them.
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• IntroductionNotationThe settingHolder continuity of intersectionsClasses of spatially independent martingalesA geometric criterion for Holder continuityAffine intersections and projectionsFractal boundaries and intersections with algebraic curvesIntersections with self-similar sets and measuresDimension of projections: applications of Theorem 4.4Upper bounds on dimensions of intersectionsLower bounds for the dimension of intersections, and dimension conservationProducts and convolutions of spatially independent martingalesApplications to Fourier decay and restrictionBibliography.
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The authors 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. The authors pair the random measures with deterministic families of parametrized measures $\{\eta_t\}_t$, and show that under some natural checkable conditions, a.s. the mass of the intersections is Holder continuous as a function of $t$. This continuity phenomenon turns out to underpin a large amount of geometric information about these measures, allowing us to unify and substantially generalize a large number of existing results on the geometry of random Cantor sets and measures, as well as obtaining many new ones.Among other things, for large classes of random fractals they establish (a) very strong versions of the Marstrand-Mattila projection and slicing results, as well as dimension conservation, (b) slicing results with respect to algebraic curves and self-similar sets, (c) smoothness of convolutions of measures, including self-convolutions, and nonempty interior for sumsets, and (d) rapid Fourier decay. Among other applications, the authors obtain an answer to a question of I. Laba in connection to the restriction problem for fractal measures.
(source: Nielsen Book Data)