"birational geometry" meaning in All languages combined

{
  "head_templates": [
    {
      "args": {
        "1": "-"
      },
      "expansion": "birational geometry (uncountable)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
          ],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Entries with translation boxes",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Pages with 1 entry",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Pages with entries",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Terms with French translations",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Algebraic geometry",
          "orig": "en:Algebraic geometry",
          "parents": [
            "Algebra",
            "Geometry",
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "examples": [
        {
          "text": "Birational geometry amounts to the study of mappings given by rational functions rather than polynomials.",
          "type": "example"
        },
        {
          "text": "1999, Yongbin Ruan, Surgery, Quantum Cohomology and Birational Geometry, Ya. Eliashberg, D. Fuchs, T. Ratiu, Alan Weinstein (editors), Northern California Symplectic Geometry Seminar, American Mathematical Society, Translations, Series 2, Volume 196, page 183,\nRecently, some amazing relations between quantum cohomology and birational geometry have been discovered."
        },
        {
          "text": "2008, Paltin Ionescu, Birational geometry of rationally connected manifolds via quasi-lines, Ciro Ciliberto, et al., Projective Varieties with Unexpected Properties, Walter de Gruyter, page 317,\nThis is, mostly, a survey of results about the birational geometry of rationally connected manifolds, using rational curves analogous to lines in ℙⁿ (quasi-lines)."
        },
        {
          "text": "2009, Claudio Bartocci, Ugo Bruzzo, Daniel Hernández Ruipérez, Fourier-Mukai and Nahm Transforms in Geometry and Mathematical Physics, Springer (Birkhäuser), Progress in Mathematics 276, page 233,\nIn this chapter we offer some applications of Fourier-Mukai transforms, namely, a classification of the Fourier-Mukai partners of complex projective surfaces, some issues in birational geometry, and an approach to the McKay correspondence via Fourier-Mukai transform."
        }
      ],
      "glosses": [
        "A field of algebraic geometry in which the aim is to determine under what conditions two algebraic varieties are isomorphic outside lower-dimensional subsets."
      ],
      "id": "en-birational_geometry-en-noun-jh9OCWmj",
      "links": [
        [
          "algebraic geometry",
          "algebraic geometry"
        ],
        [
          "algebraic varieties",
          "algebraic variety"
        ],
        [
          "isomorphic",
          "isomorphic"
        ]
      ],
      "raw_glosses": [
        "(algebraic geometry) A field of algebraic geometry in which the aim is to determine under what conditions two algebraic varieties are isomorphic outside lower-dimensional subsets."
      ],
      "related": [
        {
          "word": "birational automorphism"
        },
        {
          "word": "birational automorphism group"
        },
        {
          "word": "birational equivalence"
        },
        {
          "word": "birational invariant"
        },
        {
          "word": "birational mapping"
        },
        {
          "word": "birational morphism"
        },
        {
          "word": "blowup"
        },
        {
          "word": "plurigenus"
        }
      ],
      "tags": [
        "uncountable"
      ],
      "topics": [
        "algebraic-geometry",
        "geometry",
        "mathematics",
        "sciences"
      ],
      "translations": [
        {
          "code": "fr",
          "lang": "French",
          "sense": "field of algebraic geometry",
          "tags": [
            "feminine"
          ],
          "word": "géométrie birationnelle"
        }
      ],
      "wikipedia": [
        "birational geometry"
      ]
    }
  ],
  "word": "birational geometry"
}

{
  "head_templates": [
    {
      "args": {
        "1": "-"
      },
      "expansion": "birational geometry (uncountable)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "birational automorphism"
    },
    {
      "word": "birational automorphism group"
    },
    {
      "word": "birational equivalence"
    },
    {
      "word": "birational invariant"
    },
    {
      "word": "birational mapping"
    },
    {
      "word": "birational morphism"
    },
    {
      "word": "blowup"
    },
    {
      "word": "plurigenus"
    }
  ],
  "senses": [
    {
      "categories": [
        "English entries with incorrect language header",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with usage examples",
        "English uncountable nouns",
        "Entries with translation boxes",
        "Pages with 1 entry",
        "Pages with entries",
        "Terms with French translations",
        "en:Algebraic geometry"
      ],
      "examples": [
        {
          "text": "Birational geometry amounts to the study of mappings given by rational functions rather than polynomials.",
          "type": "example"
        },
        {
          "text": "1999, Yongbin Ruan, Surgery, Quantum Cohomology and Birational Geometry, Ya. Eliashberg, D. Fuchs, T. Ratiu, Alan Weinstein (editors), Northern California Symplectic Geometry Seminar, American Mathematical Society, Translations, Series 2, Volume 196, page 183,\nRecently, some amazing relations between quantum cohomology and birational geometry have been discovered."
        },
        {
          "text": "2008, Paltin Ionescu, Birational geometry of rationally connected manifolds via quasi-lines, Ciro Ciliberto, et al., Projective Varieties with Unexpected Properties, Walter de Gruyter, page 317,\nThis is, mostly, a survey of results about the birational geometry of rationally connected manifolds, using rational curves analogous to lines in ℙⁿ (quasi-lines)."
        },
        {
          "text": "2009, Claudio Bartocci, Ugo Bruzzo, Daniel Hernández Ruipérez, Fourier-Mukai and Nahm Transforms in Geometry and Mathematical Physics, Springer (Birkhäuser), Progress in Mathematics 276, page 233,\nIn this chapter we offer some applications of Fourier-Mukai transforms, namely, a classification of the Fourier-Mukai partners of complex projective surfaces, some issues in birational geometry, and an approach to the McKay correspondence via Fourier-Mukai transform."
        }
      ],
      "glosses": [
        "A field of algebraic geometry in which the aim is to determine under what conditions two algebraic varieties are isomorphic outside lower-dimensional subsets."
      ],
      "links": [
        [
          "algebraic geometry",
          "algebraic geometry"
        ],
        [
          "algebraic varieties",
          "algebraic variety"
        ],
        [
          "isomorphic",
          "isomorphic"
        ]
      ],
      "raw_glosses": [
        "(algebraic geometry) A field of algebraic geometry in which the aim is to determine under what conditions two algebraic varieties are isomorphic outside lower-dimensional subsets."
      ],
      "tags": [
        "uncountable"
      ],
      "topics": [
        "algebraic-geometry",
        "geometry",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "birational geometry"
      ]
    }
  ],
  "translations": [
    {
      "code": "fr",
      "lang": "French",
      "sense": "field of algebraic geometry",
      "tags": [
        "feminine"
      ],
      "word": "géométrie birationnelle"
    }
  ],
  "word": "birational geometry"
}