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This textbook uses examples, exercises, diagrams, and unambiguous proof, to help students make the link between classical and differential geometries.
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Schubert varieties lie at the cross roads of algebraic geometry, combinatorics, commutative algebra, and representation theory. They are an important class of subvarieties of flag varieties, interesting in their own right, and providing an inductive tool for studying flag varieties. The literature on them is vast, for they are ubiquitous—they have been intensively studied over the last fifty years, from many different points of view and by many different authors. This book is mainly a detailed account of a particularly interesting instance of their occurrence: namely, in relation to classical invariant theory. More precisely, it is about the connection between the first and second fundamental theorems of classical invariant theory on the one hand and standard monomial theory for Schubert varieties in certain special flag varieties - the ordinary, orthogonal, and symplectic Grassmannians - on the other. Historically, this connection was the prime motivation for the development of standard monomial theory. Determinantal varieties and basic concepts of geometric invariant theory arise naturally in establishing the connection. The book also treats, in the last chapter, some other applications of standard monomial theory, e.g., to the study of certain naturally occurring affine algebraic varieties that, like determinantal varieties, can be realized as open parts of Schubert varieties.
Schubert varieties. --- Geometry, Algebraic. --- Algebraic geometry --- Geometry --- Geometry, Algebraic --- Geometry, algebraic. --- Algebra. --- Algebraic Geometry. --- Mathematics --- Mathematical analysis --- Algebraic geometry.
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La théorie classique des suites de Sturm fournit un algorithme pour déterminer le nombre de racines d'un polynôme à coefficients réels contenues dans un intervalle donné. L'objet principal de ce mémoire est de montrer qu'une généralisation adéquate de la théorie des suites de Sturm fournit entre autres choses: une notion d'indice de Maslov pour un lacet algébrique de lagrangiens défini sur un anneau commutatif; une démonstration du théorème fondamental de la K-théorie (algébrique) hermitienne, théorème dû à M. Karoubi; une démonstration des théorèmes de périodicité de Bott (topologique), dans l'esprit des travaux de F. Latour; un calcul du groupe K2 relatif, symplectique-linéaire, pour tous les anneaux commutatifs, dans l'esprit des travaux de R. Sharpe. Le livre est dans la mesure du possible « self-contained » et élémentaire: il met essentiellement en oeuvre des arguments d'algèbre linéaire ou bilinéaire. Il présente une approche unifiée de l'indice de Maslov en termes de suites de Sturm et de formes quadratiques.
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Issu d’un cours de maîtrise de l’Université Paris VII, ce texte est réédité tel qu’il était paru en 1978. A propos du théorème de Bézout sont introduits divers outils nécessaires au développement de la notion de multiplicité d’intersection de deux courbes algébriques dans le plan projectif complexe. Partant des notions élémentaires sur les sous-ensembles algébriques affines et projectifs, on définit les multiplicités d’intersection et interprète leur somme entermes du résultant de deux polynômes. L’étude locale est prétexte à l’introduction des anneaux de série formelles ou convergentes ; elle culmine dans le théorème de Puiseux dont la convergence est ramenée par des éclatements à celle du théorème des fonctions implicites. Diverses figures éclairent le texte: on y "voit" en particulier que l’équation homogène x3+y3+z3 = 0 définit un tore dans le plan projectif complexe.
Curves, Algebraic. --- Geometry, Algebraic. --- Geometry, Plane. --- Curves, Algebraic --- Geometry, Plane --- Geometry, Algebraic --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Mathematics. --- Algebraic geometry. --- Algebraic Geometry. --- Algebraic geometry --- Plane geometry --- Algebraic curves --- Algebraic varieties --- Geometry, algebraic.
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Aimed primarily at graduate students and beginning researchers, this book provides an introduction to algebraic geometry that is particularly suitable for those with no previous contact with the subject and assumes only the standard background of undergraduate algebra. It is developed from a masters course given at the Université Paris-Sud, Orsay, and focusses on projective algebraic geometry over an algebraically closed base field. The book starts with easily-formulated problems with non-trivial solutions – for example, Bézout’s theorem and the problem of rational curves – and uses these problems to introduce the fundamental tools of modern algebraic geometry: dimension; singularities; sheaves; varieties; and cohomology. The treatment uses as little commutative algebra as possible by quoting without proof (or proving only in special cases) theorems whose proof is not necessary in practice, the priority being to develop an understanding of the phenomena rather than a mastery of the technique. A range of exercises is provided for each topic discussed, and a selection of problems and exam papers are collected in an appendix to provide material for further study.
Geometry, Algebraic --- Geometry. --- Mathematics --- Euclid's Elements --- Algebraic geometry --- Geometry --- Geometry, algebraic. --- Algebra. --- Mathematics. --- Algebraic Geometry. --- General Algebraic Systems. --- Mathematics, general. --- Math --- Science --- Mathematical analysis --- Algebraic geometry.
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Field Arithmetic explores Diophantine fields through their absolute Galois groups. This largely self-contained treatment starts with techniques from algebraic geometry, number theory, and profinite groups. Graduate students can effectively learn generalizations of finite field ideas. We use Haar measure on the absolute Galois group to replace counting arguments. New Chebotarev density variants interpret diophantine properties. Here we have the only complete treatment of Galois stratifications, used by Denef and Loeser, et al, to study Chow motives of Diophantine statements. Progress from the first edition starts by characterizing the finite-field like P(seudo)A(lgebraically)C(losed) fields. We once believed PAC fields were rare. Now we know they include valuable Galois extensions of the rationals that present its absolute Galois group through known groups. PAC fields have projective absolute Galois group. Those that are Hilbertian are characterized by this group being pro-free. These last decade results are tools for studying fields by their relation to those with projective absolute group. There are still mysterious problems to guide a new generation: Is the solvable closure of the rationals PAC; and do projective Hilbertian fields have pro-free absolute Galois group (includes Shafarevich's conjecture)? The third edition improves the second edition in two ways: First it removes many typos and mathematical inaccuracies that occur in the second edition (in particular in the references). Secondly, the third edition reports on five open problems (out of thirtyfour open problems of the second edition) that have been partially or fully solved since that edition appeared in 2005.
Algebraic fields. --- Algebraic number theory. --- Number theory --- Algebraic number fields --- Algebraic numbers --- Fields, Algebraic --- Algebra, Abstract --- Algebraic number theory --- Rings (Algebra)
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