birational geometry

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birational geometry

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birational geometry (uncountable)

1. (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.

   Birational geometry amounts to the study of mappings given by rational functions rather than polynomials.

   • 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,

     Recently, some amazing relations between quantum cohomology and birational geometry have been discovered.

   • 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,

     This is, mostly, a survey of results about the birational geometry of rationally connected manifolds, using rational curves analogous to lines in ${\displaystyle \mathbb {P} ^{n}}$ (quasi-lines).

   • 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,

     In 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.

• birational automorphism
• birational automorphism group
• birational equivalence
• birational invariant
• birational mapping
• birational morphism

• blowup
• plurigenus

• Abundance conjecture on Wikipedia.Wikipedia
• Blowing up on Wikipedia.Wikipedia
• Birational geometry on Encyclopedia of Mathematics