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Hodge theory. --- Hodge, Théorie de --- Geometrie algebrique --- Topologie algebrique --- Espaces fibres --- Geometrie algebrique --- Topologie algebrique --- Espaces fibres

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Global differential geometry --- Géométrie différentielle globale --- Connections (Mathematics) --- Global differential geometry. --- Géométrie différentielle globale. --- Courbes algébriques --- Geometrie algebrique --- Courbes algebriques

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This textbook, for an undergraduate course in modern algebraic geometry, recognizes that the typical undergraduate curriculum contains a great deal of analysis and, by contrast, little algebra. Because of this imbalance, it seems most natural to present algebraic geometry by highlighting the way it connects algebra and analysis; the average student will probably be more familiar and more comfortable with the analytic component. The book therefore focuses on Serre's GAGA theorem, which perhaps best encapsulates the link between algebra and analysis. GAGA provides the unifying theme of the book: we develop enough of the modern machinery of algebraic geometry to be able to give an essentially complete proof, at a level accessible to undergraduates throughout. The book is based on a course which the author has taught, twice, at the Australian National University.

Geometry, Algebraic --- Geometry, Analytic --- Géométrie algébrique --- Géométrie analytique --- Géométrie algébrique --- Géométrie analytique --- Geometry, Algebraic. --- Geometry, Analytic. --- Analytical geometry --- Algebra --- Conic sections --- Algebraic geometry --- Geometry --- Graphic methods

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Cet ouvrage est consacré à l'arithmétique des surfaces fibrées en courbes de genre 1 au-dessus de la droite projective, et à l'arithmétique des intersections de deux quadriques dans l'espace projectif. Swinnerton-Dyer introduisit en 1993 une technique permettant d'étudier les points rationnels des pinceaux de courbes de genre 1. La première moitié de l'ouvrage reprend et développe cette technique ainsi que ses généralisations ultérieures. La seconde moitié, qui repose sur la première, porte sur les surfaces de del Pezzo de degré 4 et sur les intersections de deux quadriques de dimension supérieure; les résultats annoncés dans [C. R. Math. Acad. Sci. Paris 342 (2006), no. 4, 223--227] y sont démontrés. This research monograph focuses on the arithmetic, over number fields, of surfaces fibred into curves of genus 1 over the projective line, and of intersections of two quadrics in projective space. The first half contains a complete account of the technique initiated by Swinnerton-Dyer in 1993 for studying rational points on pencils of curves of genus 1, while incorporating and generalising most of its subsequent refinements. The second half, which builds upon the first, is devoted to quartic del Pezzo surfaces and higher-dimensional intersections of two quadrics. It culminates in the proof of the results announced in [C. R. Math. Acad. Sci. Paris 342 (2006), no. 4, 223--227].

Algebraïsche meetkunde. --- Diofantische analyse. --- Quartic surfaces. --- Quadrics. --- Matrix pencils. --- Geometry, Algebraic. --- Surfaces quartiques --- Géométrie algébrique --- Electronic books. -- local. --- Geometry. --- Algebra --- Geometry --- Mathematics --- Physical Sciences & Mathematics --- Algebraic geometry --- Mathematics. --- Algebraic geometry. --- Number theory. --- Number Theory. --- Algebraic Geometry. --- Number study --- Numbers, Theory of --- Math --- Science --- Euclid's Elements --- Geometry, algebraic.