Algebraic Geometry

The algebraic geometry is a field of the Mathématiques to the meeting of the Géométrie and Algèbre (the commutative Algèbre in all exactitude). Basiquement, it is the study of the algebraic varieties, of the whole of points defined by polynomial equations. It is more generally interested in the Schéma S.

History

The first work concerned with the algebraic geometry goes back to the Mathématiques Arabic. Omar Khayyam proposed a method of resolution of the cubic equations by intersection of a circle and a parabola. It combined the functional Trigonométrie and approximations to obtain geometrical methods of resolution of the algebraic equations.

The “ Geometry ” of Descartes, inaugurating the study of the algebraic curved , mark the second great stage in the genesis of this discipline.

Strictly speaking, it is necessary to await the beginning of the twentieth century so that the algebraic geometry is born like part of the Géométrie with whole share. Its beginning was initiated by the Italian school of the end of the 19th century (Enriques, Chisini, Castelnuovo, Segrè…). These geometricians studied curved and surfaces of projective space (real and complex). They introduced the concepts of close points and close points in order to have a geometrical interpretation of the Théorème of Bezout. The rather free style of the Italian school remains far away from the current rigor.

See also work of max Noether in Germany.

After 1930, American schools (Zariski, Mumford…) and Frenchwomen (Weil, Samuel, Chevalley, Greenhouse…) in an algebraic form the studies of the varieties developed on a commutative body unspecified by primarily using the Théorie of the rings.

In the Années 1950 it was completely transformed by work of the French school under the impulse of Alexander Grothendieck.

In one decade, the field developed, answering traditional questions about the geometry of the algebraic varieties. Applications were very quickly given to the Théorie of the numbers. Jean-Pierre Tightens and Grothendieck provided the foundations of the theory of the beams. Towards 1960, the concept of diagram was essential.

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