Algebraic Geometry: A Concise Dictionary by Elena Rubei

By Elena Rubei

Algebraic geometry has a sophisticated, tough language. This e-book includes a definition, numerous references and the statements of the most theorems (without proofs) for each of the commonest phrases during this topic. a few phrases of similar topics are incorporated. It is helping newbies that comprehend a few, yet no longer all, simple evidence of algebraic geometry to keep on with seminars and to learn papers. The dictionary shape makes it effortless and fast to consult.

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Example text

Cone, tangent -. ([104], [228]). Let ???? be an affine algebraic variety over an algebraically closed field and let ???? = ????(????) be the ideal of ???? (see “Varieties, algebraic -, Zariski topology, regular and rational functions, morphisms and rational maps”). Let 32 | Connections ???? ∈ ????. Choose coordinates such that ???? = 0. For every ???? ∈ ????, let ????????(????) be the sum of the monomials of ???? of the lowest degree. Let ????????(????) := {????????(????)| ???? ∈ ????}. The tangent cone to ???? at ???? = 0 is defined to be the zero locus of ????????(????).

In particular, if ???? is a bundle on the projective space and ????????(????) = ????, then ????1 (????(????)) = ????1 (????) + ???? ????, where ????(????) = ???? ⊗ ???????? , where ???? is the hyperplane bundle (see “Hyperplane bundles, twisting sheaves”). The definition of Chern classes can be given, more generally, for complex bundles on compact ????∞ manifolds in a way analogous to the definition above (the only difference being in the normalization axiom). We skip it, but we mention other ways to define Chern classes. If ???? is a complex vector bundle on a compact ????∞ -manifold and ∇ is a connection on ???? (see “Connections”), we can define for any ???? ???????? (????) = ???????? (∧???? ( ???? )) , 2???????? where ???? is the curvature of ∇, ???????? is the trace, and ???? is the imaginary unit.

For some ???? and some ideal ????. In particular, if ???? is a smooth point of an algebraic variety ???? of dimension ???? over a field ????, then the completion of O????,???? is isomorphic to ????[[????1 , . . , ???????? ]]. See “Regular rings, smooth points, singular points”. Complexes. Let ???? be a ring. A complex of ????-modules, which is usually written ⋅⋅⋅ ????????−2 G ????????−1 ????????−1 G ???????? ???????? G ????????+1 ????????+1 G ⋅⋅⋅ , is the datum of a sequence of ????-modules ???????? and ????-homomorphisms ???????? : ???????? → ????????+1 such that ????????+1 ∘ ???????? = 0 for any ????.

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