...d if and only if it is isomorphic to a smooth algebraic subvariety of some complex projective space (Kodaira's projective imbedding theorem).
...D> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , '''1''' , Wiley (1978) {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR></tab
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...rojective line $\mathbf P^1$. A complete singular curve $X$ is rational if and only if its geometric genus $g$ is zero, that is, when there are no regular
When $k$ is the field $\mathbf C$ of complex numbers, the (only) non-singular complete rational curve $X$ is the Riemann
1 KB (191 words) - 10:10, 2 November 2014
One of the principal objects of study in algebraic geometry. The
modern definition of an algebraic variety as a reduced
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A generalization of the concept of a compact complex algebraic variety. A separated variety $X$ is called complete if for any variety $Y$
...tilde X$ (Nagata's theorem). A generalization of the concept of a complete algebraic variety to the relative case is that of a [[proper morphism]] of schemes.
1 KB (231 words) - 19:12, 24 November 2023
...{i\in I} a_i m_i = 0$. Thus there arises the module of syzygies, the chain complex of syzygies, etc. See also [[Hilbert syzygy theorem]].
...als and the theory of regular algebras and regular sequences, cf. [[Koszul complex]]; [[Depth of a module]].
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...is the following: A number $\alpha$ is a root of the polynomial $f(x)$ if and only if $f(x)$ is divisible by the binomial $x-\alpha$ without remainder.
...orem is called after E. Bezout [[#References|[1]]], who studied systems of algebraic equations of higher degrees.
2 KB (268 words) - 15:02, 14 February 2020
...e point, also called a node or crunode; the cusp, or spinode; the tacnode; and the ramphoid cusp.
The terms "crunode" and "spinode" are seldom used nowadays (2000).
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...proof uses only techniques from [[Algebraic topology|algebraic topology]] and so could be considered elementary. They introduced the transfer into fibre
...TD> <TD valign="top"> D.G. Quillen, "Some remarks on etale homotopy theory and a conjecture of Adams" ''Topology'' , '''7''' (1968) pp. 111–116 {{MR|022
2 KB (351 words) - 14:26, 19 April 2014
...f this complex is said to be the $n$-th [[singular homology]] group of $X$ and is denoted by $H_n(X)$.
The concept of the cohomology of a cochain complex is defined in a dual manner.
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...connected (for example, if $G$ is a Lie group), then $G^0$ is open in $G$ and $G/G^0$ is discrete.
...th the irreducible components. For every polynomial homomorphism $\phi$ of algebraic groups one has $\phi(G^0)=\phi(G)^0$. If $G$ is defined over a field, then
3 KB (419 words) - 11:16, 18 October 2014
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A numerical invariant of a one-dimensional [[Algebraic variety|algebraic variety]] defined over a field $ k $.
3 KB (483 words) - 06:28, 31 March 2023
...implex of sufficiently high dimension. The dimension of a finite geometric complex is the largest dimension of its constituent simplices.
...upper bound of the dimensions of its simplices. For an infinite geometric complex, the topology of the polyhedron induced by its imbedding into the ambient H
3 KB (419 words) - 17:16, 7 February 2011
A [[characteristic class]] of a complex bundle $\zeta$, equal to
...iplicative sequence]] corresponding to the [[power series]] $t/(1-e^{-t})$ and $c_i$ are the [[Chern class]]es.
919 bytes (125 words) - 07:42, 18 March 2023
...l plane of the torus, and the boundaries of the circles lying on the torus and obtained by its rotation from the given circle are said to be meridians of
is the radius of the axial circle, and $ \epsilon = a/l $
4 KB (589 words) - 16:43, 17 December 2019
$#C+1 = 25 : ~/encyclopedia/old_files/data/P075/P.0705170 Projective algebraic set
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3 KB (364 words) - 08:09, 13 July 2022
...tructions, which renders the algebraic space a natural object of algebraic geometry.
...f schemes becomes identical with a complete subcategory of the category of algebraic spaces.
4 KB (591 words) - 20:16, 9 October 2017
...ncyclopedia/old_files/data/I052/I.0502010 Intersection index (in algebraic geometry)
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2 KB (289 words) - 06:42, 29 December 2021
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A numerical birational invariant of a two-dimensional algebraic variety defined over an algebraically closed field $ k $.
2 KB (319 words) - 14:59, 30 March 2023
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An imbedding of an algebraic variety $ X $
5 KB (747 words) - 06:29, 30 May 2020
The classical polylogarithms are special functions of a complex variable $z$, defined by the series
for $z$ a complex number inside the unit disk and $n$ an integer at least equal to $1$. These functions satisfy the different
2 KB (237 words) - 19:07, 1 May 2023
A generalization of the concept of a non-singular [[Algebraic variety|algebraic variety]]. A [[Scheme|scheme]] $ X $
with its algebraic closure $ \overline{k} $
3 KB (540 words) - 11:38, 12 October 2023
$#C+1 = 1 : ~/encyclopedia/old_files/data/A010/A.0100450 Abstract algebraic geometry
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8 KB (1,165 words) - 16:08, 1 April 2020
...re on analytically hyperbolic varieties|range of conjectures]] linking the geometry of varieties to their Diophantine properties.
...ariski closure]] of the union of the images of all non-trivial maps from [[algebraic group]]s into $V$. Lang conjectured that the complement of the special set
2 KB (277 words) - 18:16, 2 October 2023
algebraic subvarieties of $A^n$. If $X$ is an affine algebraic variety
(see [[Affine algebraic set|Affine algebraic set]]) in $A^n$, the [[induced topology]] on $X$ is also known as the Zaris
2 KB (304 words) - 20:50, 15 March 2023
.../encyclopedia/old_files/data/A012/A.0102450 Analytic surface (in algebraic geometry)
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8 KB (1,169 words) - 18:47, 5 April 2020
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...are topologically equivalent to a circle in the Euclidean plane, while the complex projective straight line is topologically equivalent to a two-dimensional s
2 KB (335 words) - 08:23, 6 June 2020
...index ([[signature]]) $\sigma(M)$ of a compact [[Kähler manifold]] $M$ of complex dimension $2n$ can be computed by the formula
...or]]). This was proved by W.V.D. Hodge [[#References|[2]]] for the de Rham complex
3 KB (413 words) - 08:52, 23 November 2023
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...ut the importance of this invariant in the theory of the classification of algebraic varieties.
5 KB (777 words) - 22:14, 5 June 2020
$#C+1 = 27 : ~/encyclopedia/old_files/data/A011/A.0101720 Algebraic varieties, arithmetic of,
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5 KB (793 words) - 16:10, 1 April 2020
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...f a complex Kähler manifold, and whose geometry is strongly related to the geometry of this manifold.
6 KB (953 words) - 12:29, 29 December 2021
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is the field of complex numbers, the analytic bundle is said to be holomorphic). The number $ n $
6 KB (951 words) - 06:45, 22 February 2022
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of an [[Algebraic variety|algebraic variety]] $ X $,
8 KB (1,076 words) - 07:58, 21 January 2024
...sisting of a single variety implies resolution of the singularities of $V$ and can be obtained from the local uniformization theorem in dimension $\leq3$.
...">[3]</TD> <TD valign="top"> W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , '''3''' , Cambridge Univ. Press (1954) {{MR|0061846}} {{ZBL|0055.38705}}
3 KB (440 words) - 14:25, 1 May 2014
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...this concept reduces to the concept of a regular mapping (a submersion) of complex manifolds. A finitely-represented (local) morphism of schemes $ f: X \ri
3 KB (548 words) - 06:26, 21 April 2024
...cal problem of the rationality or uni-rationality of the moduli variety of algebraic curves of genus $g$.
...an algebraically closed field $k$ has the structure of a quasi-projective algebraic variety $M_g$ (see [[#References|[3]]]–[[#References|[5]]]).
5 KB (777 words) - 22:00, 7 July 2014
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as a compact algebraic variety with the aid of Grassmann coordinates (cf. [[Exterior algebra|Exter
7 KB (1,104 words) - 19:42, 5 June 2020
A (real or complex) number having the property that any number does not change if zero is adde
...y be zero. Such elements are called [[zero divisor]]s. Fields, skew-fields and [[integral domain]]s do not have zero divisors.
3 KB (471 words) - 21:06, 3 January 2015
If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.
A [[Kähler metric|Kähler metric]] on a [[Complex manifold|complex manifold]] (or orbifold) whose [[Ricci tensor|Ricci tensor]] $\operatorname
6 KB (832 words) - 16:09, 27 January 2024
...orem]]: Let $M$ and $N$ be two [[Differentiable manifold|$C^r$ manifolds]] and $f:M\to N$ a surjective $C^r$ map. If $r> \max \{0, \dim M - \dim N\}$, th
...also used. It refers to a "general" set of conditions which is consistent and is also such that every sentence from a specified set of sentences is decid
4 KB (704 words) - 11:07, 6 September 2013
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over the field of real or complex numbers is a mapping $ x \rightarrow x ^ {*} $
4 KB (550 words) - 22:13, 5 June 2020
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and $ N $
6 KB (927 words) - 22:11, 5 June 2020
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A [[Hermitian metric|Hermitian metric]] on a [[Complex manifold|complex manifold]] whose fundamental form $ \omega $
4 KB (543 words) - 22:15, 5 June 2020
of sections; and
...sheaves. If $\cF$ and $\cS$ are coherent, then so are $\cF\otimes_\cO \cS$ and ${\rm Hom}_\cO(\cF,\cS)$
3 KB (497 words) - 13:34, 6 January 2022
...(over a field $k$) all irreducible components of which have dimension $n$, and which is defined by a homogeneous ideal $I$ in the ring $k[T_0,\dots,T_N]$,
...$X$ is a non-singular connected variety, and $k=\mathbf C$ is the field of complex numbers, then
2 KB (416 words) - 12:41, 19 August 2014
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of integers, the ring of integers of some algebraic number field, the field $ \mathbf R $
4 KB (512 words) - 10:59, 29 May 2020
A complex (sometimes, real) number that is a root of a polynomial
...ynomial $x^2+1$, while $2^{1/n}$, where $n$ is any positive integer, is an algebraic number of degree $n$, being a root of the irreducible polynomial $x^n-2$.
10 KB (1,645 words) - 17:08, 14 February 2020
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sheeted covering surface over the Riemann sphere, extended over all finite and infinitely-distant branch points of $ S $.
1 KB (165 words) - 06:29, 30 May 2020
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A semi-algebraic set in $ \mathbf R ^ {n} $ (or $ k ^ {n} $,
4 KB (573 words) - 18:05, 26 January 2022
...encyclopedia/old_files/data/I050/I.0500320 Implicit function (in algebraic geometry)
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5 KB (768 words) - 22:11, 5 June 2020
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''of an algebraic variety''
3 KB (388 words) - 17:12, 7 February 2021