@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr:-ZGXHSTNB-1
  skos:prefLabel "algebraic variety"@en, "variété algébrique"@fr ;
  a skos:Concept ;
  skos:narrower psr:-BSVW2471-5 .

psr: a skos:ConceptScheme .
psr:-BSVW2471-5
  skos:inScheme psr: ;
  dc:modified "2024-10-18"^^xsd:date ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Projective_variety>, <https://fr.wikipedia.org/wiki/Vari%C3%A9t%C3%A9_projective> ;
  a skos:Concept ;
  skos:definition """En géométrie algébrique, les variétés projectives forment une classe importante de variétés. Elles vérifient des propriétés de compacité et des propriétés de finitude. C'est l'objet central de la géométrie algébrique globale. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Vari%C3%A9t%C3%A9_projective">https://fr.wikipedia.org/wiki/Vari%C3%A9t%C3%A9_projective</a>)"""@fr, """In algebraic geometry, a <b>projective variety</b> over an algebraically closed field <i>k</i> is a subset of some projective <i>n</i>-space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\mathbb {P} ^{n}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msup>           <mrow class="MJX-TeXAtom-ORD">             <mi mathvariant="double-struck">P</mi>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msup>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbb {P} ^{n}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e712abd7ff167cfb12ffa7c8c9bf7b0029cf192" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.639ex; height:2.343ex;" alt="{\\\\displaystyle \\\\mathbb {P} ^{n}}"></span> over <i>k</i> that is the zero-locus of some finite family of homogeneous polynomials of <i>n</i> + 1 variables with coefficients in <i>k</i>, that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\mathbb {P} ^{n}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msup>           <mrow class="MJX-TeXAtom-ORD">             <mi mathvariant="double-struck">P</mi>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msup>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mathbb {P} ^{n}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e712abd7ff167cfb12ffa7c8c9bf7b0029cf192" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.639ex; height:2.343ex;" alt="{\\\\displaystyle \\\\mathbb {P} ^{n}}"></span>. A projective variety is a <b>projective curve</b> if its dimension is one; it is a <b>projective surface</b> if its dimension is two; it is a <b>projective hypersurface</b> if its dimension is one less than the dimension of the containing projective space; in this case it is the set of zeros of a single homogeneous polynomial. If <i>X</i> is a projective variety defined by a homogeneous prime ideal <i>I</i>, then the quotient ring  <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle k[x_{0},\\\\ldots ,x_{n}]/I}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>k</mi>         <mo stretchy="false">[</mo>         <msub>           <mi>x</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>0</mn>           </mrow>         </msub>         <mo>,</mo>         <mo>…<!-- … --></mo>         <mo>,</mo>         <msub>           <mi>x</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>         <mo stretchy="false">]</mo>         <mrow class="MJX-TeXAtom-ORD">           <mo>/</mo>         </mrow>         <mi>I</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle k[x_{0},\\\\ldots ,x_{n}]/I}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d1e11fe5148d222f8e5b81674e1e20e0003abc6" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.949ex; height:2.843ex;" alt="{\\\\displaystyle k[x_{0},\\\\ldots ,x_{n}]/I}"></span></dd></dl> is called the homogeneous coordinate ring of <i>X</i>. Basic invariants of <i>X</i> such as the degree and the dimension can be read off the Hilbert polynomial of this graded ring. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Projective_variety">https://en.wikipedia.org/wiki/Projective_variety</a>)"""@en ;
  skos:prefLabel "projective variety"@en, "variété projective"@fr ;
  skos:broader psr:-ZGXHSTNB-1 .