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Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems. Classically, it studies zeros of multivariate polynomials ; the modern approach generalizes this in a few different aspects.
Galois had discovered new techniques to tell whether certain equations could have solutions or not. The symmetry of certain geometric objects was the key. Galois' work was picked up by André Weil who built algebraic geometry, a whole new language. Weil's work connected number theory, algebra, topology and geometry.
Algebraic geometry is the place where the algebra involved in solving systems of simultaneous multivariable polynomial equations meets the geometry of curves, surfaces, and higher dimensional algebraic varieties.
Hodge also wrote, with Daniel Pedoe, a three-volume work Methods of Algebraic Geometry, on classical algebraic geometry, with much concrete content – illustrating though what Élie Cartan called 'the debauch of indices' in its component notation. According to Atiyah, this was intended to update and replace H. F. Baker's Principles of Geometry.
Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry was split into two new subfields: synthetic geometry, which uses purely geometrical methods, and analytic geometry, which uses coordinates systemically. [23] Analytic geometry allows the study of curves unrelated to circles and lines.
Algebraic variety. Hypersurface; Quadric (algebraic geometry) Dimension of an algebraic variety; Hilbert's Nullstellensatz; Complete variety; Elimination theory; Gröbner basis; Projective variety; Quasiprojective variety; Canonical bundle; Complete intersection; Serre duality; Spaltenstein variety; Arithmetic genus, geometric genus, irregularity
Algebraic geometry became an autonomous subfield of geometry c. 1900, with a theorem called Hilbert's Nullstellensatz that establishes a strong correspondence between algebraic sets and ideals of polynomial rings. This led to a parallel development of algebraic geometry, and its algebraic counterpart, called commutative algebra. [106]
There are several equivalent definitions of a Riemann surface. A Riemann surface X is a connected complex manifold of complex dimension one. This means that X is a connected Hausdorff space that is endowed with an atlas of charts to the open unit disk of the complex plane: for every point x ∈ X there is a neighbourhood of x that is homeomorphic to the open unit disk of the complex plane, and ...