[[Abstract algebraic geometry]] |
[[Algebraic homotopy]] |
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...^n/k)$ is in one-to-one correspondence with the set of closed subvarieties in $P_k^n$.
...is the direct sum of the schemes $\operatorname{Hilb}^P(X/S)$ over all $P\in\mathbf Q(z)$. For any connected ground scheme $S$ the scheme $\operatorname
3 KB (484 words) - 05:48, 17 April 2024
is representable if and only if there is an object $ A \in \mathop{\rm Ob} \mathfrak R $
and an element $ a \in F ( A) $
6 KB (840 words) - 17:39, 16 July 2024
...heme. The structure sheaf of a Noetherian scheme and, in particular, of an algebraic variety is coherent.
...; and g) a theory of [[local cohomology]], useful in the study of coherent algebraic sheaves on incomplete varieties. One of the most important of its applicati
4 KB (570 words) - 15:59, 18 July 2024
''affine algebraic variety''
A generalization of the concept of an [[Affine algebraic set|affine algebraic set]]. An affine variety is a reduced [[Affine scheme|affine scheme]] $ X
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for a smooth [[algebraic variety]] $ X $
in the category $ \mathop{\rm Sch}\nolimits /S $
7 KB (1,008 words) - 05:56, 18 July 2024
A one-dimensional [[Algebraic variety|algebraic variety]], defined over an algebraically closed field $k$, whose field of r
In classic literature a rational curve is also called a unicursal curve.
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...equality between the degree and the dimension of a special divisor on an [[algebraic curve]]. It was proved by W. Clifford.
...ly closed field, and let $D$ be a divisor on $X$ (cf. [[Divisor (algebraic geometry)]]). Let $\deg D$ be the degree and $l(D)$ the dimension of $D$. A positive
2 KB (319 words) - 05:40, 9 April 2023
This page is a copy of the article [[Picard scheme]] in order to test [[User:Maximilian_Janisch/latexlist|automatic LaTeXification]
...itrary $5$-scheme $x$ one considers the relative Picard functor $PICX / S$ in the category $\operatorname { sch } / S$ of schemes over the scheme $5$. Th
7 KB (1,013 words) - 13:44, 17 October 2019
''in algebraic geometry''
...variant of cohomology type associated with schemes in [[Algebraic K-theory|algebraic $ K $-
8 KB (1,120 words) - 20:06, 31 October 2023
...on of the field of rational functions $k(\mathbf{P}^n)/k(X)$ is separable. In other words, $k(X)$ has a [[separable extension]] which is purely transcend
...h problem]]; the answer is negative in general, although all unirational [[algebraic curve]]s are rational.
1 KB (200 words) - 13:11, 25 November 2023
[[Abstract algebraic geometry]] |
[[Algebraic homotopy]] |
5 KB (477 words) - 23:07, 8 March 2012
$#C+1 = 1 : ~/encyclopedia/old_files/data/A010/A.0100450 Abstract algebraic geometry
...tants, which was developed at the same time, was of fundamental importance in this proof.
8 KB (1,166 words) - 06:19, 21 July 2024
''in algebraic geometry''
on an [[Algebraic variety|algebraic variety]] or [[Scheme|scheme]] $ X $
2 KB (317 words) - 15:53, 10 April 2023
...ic homotopy theory, and I usually deal with algebraic varieties related to algebraic groups.
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...row Y$ is a closed morphism, i.e. it maps closed subsets of $X \times Y$ (in the [[Zariski topology]]) into closed subsets of $Y$.
...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
..., cocomplete, [[Well-powered category|well-powered]], and co-well-powered. In particular, the product of a family of sets (exists and) coincides with its
...n of the category of polygons" ''Mat. Sb.'' , '''80''' (1969) pp. 492–502 (In Russian) {{MR|}} {{ZBL|0203.31403}} </TD></TR></table>
4 KB (570 words) - 21:02, 21 December 2017
...\dim X$. In such a case the geometric genus of $X$ is denoted by $p_g(X)$. In accordance with Serre's duality theorem
...onal surface|Rational surface]]) and also in the general classification of algebraic surfaces. The geometric genera of birationally-isomorphic smooth projective
2 KB (276 words) - 06:27, 31 March 2023
One of the principal objects of study in algebraic geometry. The
modern definition of an algebraic variety as a reduced
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''affine algebraic $k$-set''
The set of solutions of a given system of algebraic equations. Let $k$
4 KB (616 words) - 21:49, 30 March 2012