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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Projective variety ： ウィキペディア英語版 Projective variety In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space P''n'' over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables with coefficients in ''k'', that generate a prime ideal, the defining ideal of the variety. If the condition of generating a prime ideal is removed, such a set is called a projective algebraic set. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of P''n''. A Zariski open subvariety of a projective variety is called a quasi-projective variety. If ''X'' is a projective variety defined by a homogeneous prime ideal ''I'', then the quotient ring :$k(\backslash ldots,\; x\_n)/I$ is called the homogeneous coordinate ring of ''X''. The ring comes with the Hilbert polynomial ''P'', an important invariant (depending on embedding) of ''X''. The degree of ''P'' is the dimension ''r'' of ''X'' and its leading coefficient times r! is the degree of the variety ''X''. The arithmetic genus of ''X'' is (−1)''r'' (''P''(0) − 1) when ''X'' is smooth. For example, the homogeneous coordinate ring of P''n'' is $k(\backslash ldots,\; x\_n)$ and its Hilbert polynomial is $P(z)\; =\; \backslash binom$; its arithmetic genus is zero. Another important invariant of a projective variety ''X'' is the Picard group $\backslash operatorname(X)$ of ''X'', the set of isomorphism classes of line bundles on ''X''. It is isomorphic to $H^1(X,\; ^$ *). It is an intrinsic notion (independent of embedding). For example, the Picard group of P''n'' is isomorphic to Z via the degree map. The kernel of $\backslash operatorname:\; \backslash operatorname(X)\; \backslash to\; \backslash mathbf$ is called the Jacobian variety of ''X''. The Jacobian of a (smooth) curve plays an important role in the study of the curve. The classification program, classical and modern, naturally leads to the construction of moduli of projective varieties. A Hilbert scheme, which is a projective scheme, is used to parametrize closed subschemes of P''n'' with the prescribed Hilbert polynomial. For example, a Grassmannian $\backslash mathbb(k,\; n)$ is a Hilbert scheme with the specific Hilbert polynomial. The geometric invariant theory offers another approach. The classical approaches include the Teichmüller space and Chow varieties. For complex projective varieties, there is a marriage of algebraic and complex-analytic approaches. Chow's theorem says that a subset of the projective space is the zero-locus of a family of holomorphic functions if and only if it is the zero-locus of homogeneous polynomials. (A corollary of this is that a "compact" complex space admits at most one variety structure.) The GAGA says that the theory of holomorphic vector bundles (more generally coherent analytic sheaves) on ''X'' coincide with that of algebraic vector bundles. == Examples == *The fibered product of two projective spaces is projective. In fact, there is the explicit immersion (called Segre embedding) ::$\backslash mathbf^n\; \backslash times\; \backslash mathbf^m\; \backslash to\; \backslash mathbf^,\; (x\_i,\; y\_j)\; \backslash mapsto\; x\_iy\_j$ (lexicographical order). :It follows from this that the fibered product of projective varieties is also projective. *Every irreducible closed subset of P''n'' of codimension one is a hypersurface; i.e., the zero set of some homogeneous irreducible polynomial.〔; this is because the homogeneous coordinate ring of P''n'' is a unique factorization domain and in a UFD every prime ideal of height 1 is principal.〕 * The arithmetic genus of a hypersurface of degree ''d'' is $\backslash binom$ in $\backslash mathbf^n$. In particular, a smooth curve of degree ''d'' in P2 has arithmetic genus $(d-1)(d-2)/2$. This is the genus formula. *A smooth curve is projective if and only if it is complete. This is because of the following consideration. If ''F'' is the function field of a smooth projective curve ''C'' (called the algebraic function field), then ''C'' may be identified with the set of discrete valuation rings of ''F'' over ''k'' and this set has a natural Zariski topology called the Zariski–Riemann space. See also algebraic curve for more specific examples of curves. *A smooth complete curve of genus one is called an elliptic curve. By an argument with the Riemann-Roch theorem, one can show that such a curve can be embedded as a closed subvariety in P2. (In general, any (smooth complete) curve can be embedded in P3.) Conversely, any smooth closed curve in P2 of degree three has genus one by the genus formula and is thus an elliptic curve. An elliptic curve is isomorphic to its own Jacobian and thus an abelian variety. *A smooth complete curve of genus greater than or equal to two is called a hyperelliptic curve if there is a finite morphism $C\; \backslash to\; \backslash mathbf^1$ of degree two. *An abelian variety (i.e., a complete group variety) admits an ample line bundle and thus projective. On the other hand, an abelian scheme may not be projective. Examples of abelian varieties are elliptic curves, Jacobian varieties and K3 surfaces. *Some (but not all) complex tori are projective. A complex torus is of the form $\backslash mathbb^g/L$ (period lattice construction) as a complex Lie group where ''L'' is a lattice and ''g'' is the complex dimension of the torus. Suppose $g=1$. Let $\backslash wp$ be the Weierstrass's elliptic function. The function satisfies a certain differential equation and as a consequence it defines a closed immersion: *:$\backslash mathbb/L\; \backslash to\; \backslash mathbf^2,\; L\; \backslash mapsto\; (0:0:1),\; z\; \backslash mapsto\; (1\; :\; \backslash wp(z)\; :\; \backslash wp\text{'}(z))$ :for some lattice ''L''. Thus, $\backslash mathbb/L$ is an elliptic curve. The uniformization theorem implies that every complex elliptic curve arises in this way. The case $g\; >\; 1$ is more complicated; it is a matter of polarization. (cf. Lefschetz's embedding theorem.) By the p-adic uniformization, the case $g\; =\; 1$ has a ''p''-adic analog. *Flag varieties are projective in the natural way. *The Plücker embedding exhibits a Grassmannian as a projective variety. *(Riemann) A compact Riemann surface (i.e., compact complex manifold of dimension one) is a projective variety. By the Torelli theorem, it is uniquely determined by its Jacobian. *(Chow-Kodaira) A compact complex manifold of dimension two with two algebraically independent meromorphic functions is a projective variety. *An affine variety is almost never projective. In fact, a projective subvariety of an affine variety must have dimension zero. This is because only the constants are globally regular functions on a projective variety. *The Kodaira embedding theorem gives a criterion for a Kähler manifold to be projective. Note however that it is very hard to decide whether a complex manifold is kähler or not. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Projective variety」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.