...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
5 KB (706 words) - 13:35, 17 March 2023
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]].
906 bytes (150 words) - 16:48, 23 November 2023
...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).
1 KB (186 words) - 13:06, 23 July 2014
...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.
2 KB (265 words) - 21:51, 22 October 2016
...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