Get Algebraic geometry. A concise dictionary PDF

By Elena Rubei

ISBN-10: 3110316226

ISBN-13: 9783110316223

Algebraic geometry has a classy, tough language. This publication incorporates a definition, a number of references and the statements of the most theorems (without proofs) for each of the most typical phrases during this topic. a few phrases of similar topics are integrated. It is helping novices that comprehend a few, yet now not all, uncomplicated evidence of algebraic geometry to stick to seminars and to learn papers. The dictionary shape makes it effortless and speedy to refer to.

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Example text

107], [129], [140], [197], [201], [202], [228]). Definition. , the image through ???? of any closed subset (closed in the Zariski topology) is a closed subset. Proposition. An algebraic variety ???? over ℂ is complete if and only if it is compact with the usual topology. Proposition. A projective algebraic variety ???? over an algebraically closed field is complete. There exist complete nonprojective algebraic varieties; see [201]. Chow’s lemma. For any complete variety ???? over an algebraically closed field, there exists a projective algebraic variety ???? and a surjective birational morphism from ???? to ????.

Let ???? be an algebraic variety (or a manifold) and ???? and ???? be two vector bundles on ???? and let ???? : ???? → ???? be a morphism of vector bundles. For any ???? ∈ ℕ, the set ???????? (????) = {???? ∈ ????| ????????(???????? : ???????? → ???????? ) ≤ ????} is said to be a determinantal variety (or the ????-degeneracy locus of ????). Example. ,???? O(???????? ). Then ???? is given by a matrix ???? × ???? whose entry ????, ???? is a polynomial of degree ???????? − ???????? if ???????? ≥ ???????? and 0 if ???????? < ???????? and ???????? (????) is the zero locus in ℙ???? of the minors (???? + 1) × (???? + 1) of ????.

G???? We have that ????∗ (ℙ(????), ℤ) is a free ????∗ (????, ℤ)-module generated by 1, ????, . . , ????????−1 , where ???? = ????1 (????−1 ). Theorem. If ???? : ℙ(????) → ???? is the projectivized bundle of ???? and ???? = ????1 (????−1 ), where ???? is the tautological bundle on ℙ(????), we have ???????? + ∑ ????∗ (???????? (????))????????−???? = 0 . ???? The theorem above is sometimes used (once ????0 and ????1 are defined) to define the Chern classes (to be the unique elements satisfying the equation above). Some useful formulas about Chern classes are: (i) For any ???? ???????? (????∨ ) = (−1)???? ???????? (????).

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