Download e-book for kindle: Algebraic Geometry: A Concise Dictionary by Elena Rubei

By Elena Rubei

ISBN-10: 3110316226

ISBN-13: 9783110316223

Algebraic geometry has a sophisticated, tough language. This publication encompasses a definition, a number of references and the statements of the most theorems (without proofs) for each of the commonest phrases during this topic. a few phrases of comparable matters are integrated. It is helping newbies that comprehend a few, yet now not all, uncomplicated proof of algebraic geometry to persist with seminars and to learn papers. The dictionary shape makes it effortless and quickly to consult.

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Additional info for Algebraic Geometry: A Concise Dictionary

Example text

We recall that if ???? : ???? → ???? is a holomorphic surjective map between complex manifolds such that the differential of ???? at every point has maximal rank and the fibres of ???? are compact complex manifolds, then the fibres are diffeomorphic by Ehresmann’s theorem (see [58] or [149, Theorem. 3]), and so ???? parametrizes some complex structures on the same differentiable manifold. Definition. Let ???? be a compact complex manifold. A deformation of ???? is the datum of a proper flat morphism ???? : ???? → ???? of complex analytic spaces, a point ???? ∈ ???? and an isomorphism ????−1 (????) ≅ ????.

Let ???? be a complex manifold and ???? be a holomorphic vector bundle with a Hermitian metric. There is a unique connection on ???? compatible both with the holomorphic structure and with the metric. Covering projections | 35 Correspondences. ([72], [93], [196]). Let ???? and ???? be two algebraic varieties; a correspondence from ???? to ???? is an algebraic cycle in ???? × ???? (see “Cycles”). Let ????, ????, ???? be three smooth algebraic varieties and let ????????,???? be the projection from ???? × ???? × ???? onto ???? × ???? and analogously ????????,???? and ????????,???? .

It is not ramified, we say that it is an étale covering projection. See also “Riemann’s existence theorem” Cremona transformations. See “Quadratic transformations, Cremona transfor- mations”. 38 | Cross ratio Cross ratio. Let ???? be a field and let ℙ1 = ℙ1???? . The cross ratio of an ordered set of ???? four points of ℙ , (????1 , ????2 , ????3 , ????4 ), with ????1 , ????2 , ????3 distinct, is the element [ ???? 1 ] ∈ ℙ1 such 1 2 that there exists an automorphism ???? : ℙ1 → ℙ1 such that ???? 1 1 0 (????(????1 ), ????(????2 ), ????(????3 ), ????(????4 )) = ([ ] , [ ] , [ ] , [ 1 ]) , ????2 1 0 1 or, using not homogeneous coordinates, (????(????1 ), ????(????2 ), ????(????3 ), ????(????4 )) = (∞, 0, 1, ????), where ???? = ???? 2 /???? 1 .

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