...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).
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 $
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$#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
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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
$#C+1 = 57 : ~/encyclopedia/old_files/data/C024/C.0204160 Complex space,
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5 KB (691 words) - 17:46, 4 June 2020
The divisor class group under algebraic equivalence on a non-singular projective variety.
...ent group $D(X)/D_{\mathrm{a}}(X)$ is called the Néron–Severi group of $X$ and is denoted by $\mathrm{NS}(X)$. The Néron–Severi theorem asserts that th
4 KB (687 words) - 05:47, 15 April 2023
geometry, which reduces the problem of the existence of rational
points on an algebraic variety over a global field to the analogous
3 KB (543 words) - 15:48, 17 February 2012
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...ements points, straight lines, planes, circles, spheres, conics, quadrics, and multi-dimensional analogues of these.
8 KB (1,191 words) - 06:39, 9 April 2023
An [[Algebraic variety|algebraic variety]] which is minimal relative to the existence of birational morphism
...gularities]]) for varieties of arbitrary dimension in characteristic 0 for and for varieties of dimension $n\leq3$ in characteristic $p>5$.
7 KB (1,005 words) - 21:13, 21 November 2018
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of a finite [[CW-complex|CW-complex]] (cf. also [[Cellular space|Cellular space]]) $ X $
11 KB (1,584 words) - 11:51, 8 April 2023
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on a compact [[Complex manifold|complex manifold]] $ X $.
5 KB (719 words) - 22:14, 5 June 2020
[[Cohomology ring|cohomology ring]] of a complex flag manifold $G/P$ (cf. also
[[Linear algebraic group|linear algebraic group]] and $P$ is a
3 KB (371 words) - 18:45, 30 March 2012
...ordinates in a Cartesian coordinate system satisfy an [[Algebraic equation|algebraic equation]] of degree two:
...in the general equation (*), one may transform it by parallel translation and rotation in the coordinate system to one of the 17 canonical forms given be
19 KB (2,446 words) - 17:35, 31 March 2012
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...or finite-dimensional orientable topological manifolds [[#References|[1]]] and for finite cell complexes (see [[#References|[2]]], [[#References|[3]]]). T
10 KB (1,491 words) - 19:04, 26 March 2023
A pair $(X,\xi)$, where $V$ is a complex smooth variety (cf.
[[Algebraic variety|Algebraic variety]]) over an algebraically closed
4 KB (644 words) - 13:06, 17 April 2023
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dimensional manifold) and for which the identity imbedding $ i: N \rightarrow M $
7 KB (987 words) - 08:24, 6 June 2020
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...rmulas in algebraic geometry are the Plücker formulas for a planar reduced and irreducible curve $ Z \subset \mathbf C P ^ {2} $,
5 KB (681 words) - 08:06, 6 June 2020
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and a decomposition $ H _ {\mathbf C } = \oplus _ {p + q = n } H ^ {p,q} $
9 KB (1,281 words) - 12:14, 12 January 2021
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onto the complex plane $ \mathbf C $
7 KB (1,010 words) - 19:41, 5 June 2020
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A [[Homogeneous complex manifold|homogeneous complex manifold]] that is isomorphic to a bounded domain in $ \mathbf C ^ {n} $
5 KB (656 words) - 03:11, 4 January 2022
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...non-singular variety. More precisely, a resolution of singularities of an algebraic variety $ X $
7 KB (965 words) - 08:11, 6 June 2020
[[Algebraic group|algebraic group]] that is a complete
[[Algebraic variety|algebraic variety]]. The completeness condition
8 KB (1,216 words) - 20:39, 5 March 2012
...ive; the property of points to be on one line (collinearity), the order of algebraic curves, etc., are such properties.
...ted belong to the improper line completing this plane. All improper points and lines belong to the improper plane. By completing affine space to projectiv
7 KB (1,119 words) - 13:21, 12 December 2013
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is normal. In a neighbourhood of such a point the space has a reduced and irreducible model. Every simple (non-singular) point is normal. The simples
8 KB (1,143 words) - 16:48, 19 February 2022
''of a complete smooth algebraic variety $X$ over an algebraically
property: For any algebraic family of divisors $D$ on $X$ with base
6 KB (948 words) - 11:41, 31 January 2022
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in a complex space $ X $
7 KB (990 words) - 19:38, 5 June 2020
...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...schemes over $\mathbf{Z}$ with Hermitian complex geometry on their set of complex points. The goal is to provide a geometric framework for the study of Dioph
8 KB (1,219 words) - 21:00, 13 July 2020
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and $ T _ {Y} $
4 KB (554 words) - 06:02, 18 April 2023
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An unbounded domain in the $ ( n + m ) $-dimensional complex affine space of the form
4 KB (618 words) - 07:11, 30 December 2021
...ctive algebraic varieties without singular points over $k$, and Lie groups and their homogeneous spaces.
...ar points" . The concept of an analytic space was introduced in the 1950's and became the principal subject of the theory of analytic functions; many fund
7 KB (1,028 words) - 13:31, 25 April 2014
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...cally irreducible; for such a scheme the concepts of a connected component and an irreducible component are the same. The set of singular points of a Noet
5 KB (706 words) - 15:10, 7 June 2020
...ertion about the various objects in algebraic geometry (cohomology spaces, algebraic varieties, schemes, fibrations, etc.), stating that these objects depend on
...rst circle of finiteness theorems relates to cohomology spaces of coherent algebraic sheaves. The fundamental theorem states that these spaces are finite-dimens
11 KB (1,549 words) - 20:08, 17 December 2019
The theory of eliminating unknowns from systems of algebraic equations. More precisely, suppose one is given a system of equations
...that of finding conditions on the coefficients of a system of homogeneous algebraic equations ensuring that this system has a non-zero solution.
8 KB (1,363 words) - 16:04, 13 January 2021
$#C+1 = 183 : ~/encyclopedia/old_files/data/A011/A.0101490 Algebraic function
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20 KB (3,036 words) - 07:17, 15 June 2022
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...zed to be related to problems of [[Algebraic geometry|algebraic geometry]] and [[Representation theory|representation theory]], besides [[Spectral theory|
8 KB (1,099 words) - 08:01, 20 February 2023
$#C+1 = 126 : ~/encyclopedia/old_files/data/A011/A.0101460 Algebraic cycle
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12 KB (1,874 words) - 06:08, 17 April 2024
...ls representing geometric conditions used to solve problems in enumerative geometry. This originated in work of M. Chasles
{{Cite|Ch}} on conics and was systematized and used to great effect by H. Schubert in
8 KB (1,263 words) - 08:49, 30 March 2012
...s'''; they correspond to invertible sheaves of $ \mathcal{O}_{X} $-modules and are closely connected with [[Divisor|divisors]] on $ X $; the set of line b
...bundle $ \lambda^{n} E $ is said to be the '''determinant bundle'''. To an algebraic vector bundle $ E $, one can associate the projective bundle $ \mathbf{P}(E
14 KB (2,169 words) - 08:12, 14 December 2016
non-zero elements is not empty and forms a group under multiplication
[[Associative rings and algebras|Associative rings and algebras]]). A
6 KB (929 words) - 00:28, 18 May 2013
...encyclopedia/old_files/data/E037/E.0307070 Extension theorems (in analytic geometry)
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7 KB (1,033 words) - 19:38, 5 June 2020
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and some finite set of numbers (real or complex) by means of a finite number of arithmetical operations. A rational functio
8 KB (1,257 words) - 03:49, 4 March 2022
$#C+1 = 43 : ~/encyclopedia/old_files/data/G043/G.0403740 General\AAhtype algebraic surface,
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7 KB (986 words) - 19:41, 5 June 2020
...K$. In particular, $\operatorname{Max}T_n$ is the polydisc of unit radius, and, more generally, for arbitrary $A$ the space $\operatorname{Max}A$ is an an
...coverings by admissible sets. Morphisms of spaces with a rigid structure, and also the concept of the rigid structure induced on a subspace, are defined
6 KB (892 words) - 20:07, 22 December 2018
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An algebraic or analytic complete non-singular surface $ X $
15 KB (2,046 words) - 05:46, 13 June 2022
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...he lines that belongs to this surface are called (rectilinear) generators, and every curve that intersects all the generators is called a directrix. If $
12 KB (1,684 words) - 08:12, 6 June 2020
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on an algebraic or analytic variety $ X $
10 KB (1,385 words) - 03:10, 2 March 2022
...h of algebra whose main study is derived functors on various categories of algebraic objects (modules over a given ring, sheaves, etc.).
...is done in analytic geometry (with the difference that the transition from geometry to algebra in homology theory is irreversible).
12 KB (1,885 words) - 23:48, 23 April 2017
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is a [[Complex (in homological algebra)|complex (in homological algebra)]] $ C $:
4 KB (611 words) - 01:36, 5 March 2022
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...bra, algebraic geometry and algebraic $K$-theory; see [[#References|[a5]]] and [[#References|[a6]]].
8 KB (1,239 words) - 17:01, 1 July 2020
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be the set of all such non-splitting curves and let $ \rho $
8 KB (1,242 words) - 07:04, 6 May 2022
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is surjective, and let $ {\mathcal J} = \mathop{\rm Ker} f ^ { \# } $.
5 KB (799 words) - 08:03, 6 June 2020
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...etric is induced by the [[Fubini–Study metric|Fubini–Study metric]] on the complex projective space. Similarly, every submanifold in the affine space $ \mat
12 KB (1,750 words) - 19:20, 27 October 2023
The ''Chow ring'' of a non-singular quasi-projective algebraic variety
...of rational equivalence classes of algebraic cycles (cf. [[Algebraic cycle|Algebraic cycle]]) on that variety. Multiplication in this ring is defined in terms o
4 KB (714 words) - 21:54, 24 April 2012
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...nalogue of a [[Hyper-elliptic curve|hyper-elliptic curve]]. A non-singular algebraic projective surface $ X $
5 KB (749 words) - 07:33, 22 December 2023