See algebraic variety in All languages combined, or Wiktionary
Download JSON data for algebraic variety meaning in English (5.2kB)
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"ref": "2005, Giovanni Giachetta, Luigi Mangiarotti, Gennadi Sardanashvili, Geometric and Algebraic Topological Methods in Quantum Mechanics, World Scientific, page 87",
"text": "If #x5C;mathcal#x7B;W#x7D; is a locally closed subset (i.e., the intersection of open and closed sets) of an algebraic variety, it becomes an algebraic variety in a natural manner since the germs of regular functions at x#x5C;in#x5C;mathcal#x7B;W#x7D; axe taken to be the germs of functions on #x5C;mathcal#x7B;W#x7D; induced by functions in the stalk #x5C;mathfrak#x7B;R#x7D;#x5F;x. The definitions of irreducibility and local rings of subvarieties for algebraic varieties are given in the same manner as before. From now on, by a variety is meant an algebraic variety. Any variety (X,#x5C;mathfrak#x7B;R#x7D;), by definition, is a local-ringed space.",
"type": "quotation"
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{
"ref": "2011, Carel Faber, Gerard van der Geer, Eduard Looijenga, Classification of Algebraic Varieties, European Mathematical Society, page vii",
"text": "The quest for understanding the structure of algebraic varieties is a very natural one.[…]A landmark from that time was Hironaka's 1964 theorem on the existence of a smooth model for any complex algebraic variety.",
"type": "quotation"
},
{
"ref": "2014, Herwig Hauser, “Blowups and Resolution”, in David Ellwood, Herwig Hauser, Shigefumi Mori, Josef Schicho, editors, The Resolution of Singular Algebraic Varieties, American Mathematical Society, page 7",
"text": "Remark 2.8 Abstract algebraic varieties are obtained by gluing affine algebraic varieties along principal open subsets, cf. [Mum99] I, §3, §4, [Sha94] V, §3. This allows to develop the category of algebraic varieties with the usual constructions therein. All subsequent definitions could be formulated for abstract algebraic varieties, but will only be developed in the affine or quasi-affine case to keep things simple.",
"type": "quotation"
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"glosses": [
"The set of solutions of a given system of polynomial equations over the real or complex numbers; any of certain generalisations of such a set that preserves the geometric intuition implicit in the original definition."
],
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{
"sense": "set of solutions to a system of polynomial equations, any of certain generalisations of such a set",
"word": "affine variety"
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{
"sense": "set of solutions to a system of polynomial equations, any of certain generalisations of such a set",
"word": "algebraic manifold"
},
{
"sense": "set of solutions to a system of polynomial equations, any of certain generalisations of such a set",
"word": "projective variety"
},
{
"sense": "set of solutions to a system of polynomial equations, any of certain generalisations of such a set",
"word": "algebraic curve"
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"id": "en-algebraic_variety-en-noun-d7X~1WY7",
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"(algebraic geometry) The set of solutions of a given system of polynomial equations over the real or complex numbers; any of certain generalisations of such a set that preserves the geometric intuition implicit in the original definition."
],
"related": [
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"word": "algebraic set"
}
],
"synonyms": [
{
"english": "elliptical",
"word": "variety"
}
],
"topics": [
"algebraic-geometry",
"geometry",
"mathematics",
"sciences"
],
"translations": [
{
"code": "fi",
"lang": "Finnish",
"sense": "set of solutions to a system of polynomial equations; any of certain generalisations of such a set",
"word": "varisto"
},
{
"code": "gl",
"lang": "Galician",
"sense": "set of solutions to a system of polynomial equations; any of certain generalisations of such a set",
"tags": [
"feminine"
],
"word": "variedade alxebraica"
},
{
"code": "it",
"lang": "Italian",
"sense": "set of solutions to a system of polynomial equations; any of certain generalisations of such a set",
"tags": [
"feminine"
],
"word": "varietà algebrica"
},
{
"code": "ru",
"lang": "Russian",
"roman": "algebraíčeskoje mnogoobrázije",
"sense": "set of solutions to a system of polynomial equations; any of certain generalisations of such a set",
"tags": [
"neuter"
],
"word": "алгебраи́ческое многообра́зие"
}
],
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"word": "algebraic variety"
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"word": "affine variety"
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"sense": "set of solutions to a system of polynomial equations, any of certain generalisations of such a set",
"word": "algebraic manifold"
},
{
"sense": "set of solutions to a system of polynomial equations, any of certain generalisations of such a set",
"word": "projective variety"
},
{
"sense": "set of solutions to a system of polynomial equations, any of certain generalisations of such a set",
"word": "algebraic curve"
}
],
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"pos": "noun",
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"ref": "2005, Giovanni Giachetta, Luigi Mangiarotti, Gennadi Sardanashvili, Geometric and Algebraic Topological Methods in Quantum Mechanics, World Scientific, page 87",
"text": "If #x5C;mathcal#x7B;W#x7D; is a locally closed subset (i.e., the intersection of open and closed sets) of an algebraic variety, it becomes an algebraic variety in a natural manner since the germs of regular functions at x#x5C;in#x5C;mathcal#x7B;W#x7D; axe taken to be the germs of functions on #x5C;mathcal#x7B;W#x7D; induced by functions in the stalk #x5C;mathfrak#x7B;R#x7D;#x5F;x. The definitions of irreducibility and local rings of subvarieties for algebraic varieties are given in the same manner as before. From now on, by a variety is meant an algebraic variety. Any variety (X,#x5C;mathfrak#x7B;R#x7D;), by definition, is a local-ringed space.",
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},
{
"ref": "2011, Carel Faber, Gerard van der Geer, Eduard Looijenga, Classification of Algebraic Varieties, European Mathematical Society, page vii",
"text": "The quest for understanding the structure of algebraic varieties is a very natural one.[…]A landmark from that time was Hironaka's 1964 theorem on the existence of a smooth model for any complex algebraic variety.",
"type": "quotation"
},
{
"ref": "2014, Herwig Hauser, “Blowups and Resolution”, in David Ellwood, Herwig Hauser, Shigefumi Mori, Josef Schicho, editors, The Resolution of Singular Algebraic Varieties, American Mathematical Society, page 7",
"text": "Remark 2.8 Abstract algebraic varieties are obtained by gluing affine algebraic varieties along principal open subsets, cf. [Mum99] I, §3, §4, [Sha94] V, §3. This allows to develop the category of algebraic varieties with the usual constructions therein. All subsequent definitions could be formulated for abstract algebraic varieties, but will only be developed in the affine or quasi-affine case to keep things simple.",
"type": "quotation"
}
],
"glosses": [
"The set of solutions of a given system of polynomial equations over the real or complex numbers; any of certain generalisations of such a set that preserves the geometric intuition implicit in the original definition."
],
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"(algebraic geometry) The set of solutions of a given system of polynomial equations over the real or complex numbers; any of certain generalisations of such a set that preserves the geometric intuition implicit in the original definition."
],
"synonyms": [
{
"english": "elliptical",
"word": "variety"
}
],
"topics": [
"algebraic-geometry",
"geometry",
"mathematics",
"sciences"
],
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"translations": [
{
"code": "fi",
"lang": "Finnish",
"sense": "set of solutions to a system of polynomial equations; any of certain generalisations of such a set",
"word": "varisto"
},
{
"code": "gl",
"lang": "Galician",
"sense": "set of solutions to a system of polynomial equations; any of certain generalisations of such a set",
"tags": [
"feminine"
],
"word": "variedade alxebraica"
},
{
"code": "it",
"lang": "Italian",
"sense": "set of solutions to a system of polynomial equations; any of certain generalisations of such a set",
"tags": [
"feminine"
],
"word": "varietà algebrica"
},
{
"code": "ru",
"lang": "Russian",
"roman": "algebraíčeskoje mnogoobrázije",
"sense": "set of solutions to a system of polynomial equations; any of certain generalisations of such a set",
"tags": [
"neuter"
],
"word": "алгебраи́ческое многообра́зие"
}
],
"word": "algebraic variety"
}
This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2024-04-26 from the enwiktionary dump dated 2024-04-21 using wiktextract (93a6c53 and 21a9316). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
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