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- In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic curve such that almost all fibers are smooth curves of genus 1. (Over an algebraically closed field such as the complex numbers, these fibers are elliptic curves, perhaps without a chosen origin.) This is equivalent to the generic fiber being a smooth curve of genus one. This follows from proper base change. The surface and the base curve are assumed to be non-singular (complex manifolds or regular schemes, depending on the context). The fibers that are not elliptic curves are called the singular fibers and were classified by Kunihiko Kodaira. Both elliptic and singular fibers are important in string theory, especially in F-theory. Elliptic surfaces form a large class of surfaces that contains many of the interesting examples of surfaces, and are relatively well understood in the theories of complex manifolds and smooth 4-manifolds. They are similar to (have analogies with, that is), elliptic curves over number fields. (en)
- 数学では、楕円曲面（だえんきょくめん、英: elliptic surface）は楕円ファイバーを持つ曲面であり、言い換えると、曲面からの代数曲線への連結な固有射が、ほとんどの点上のファイバーを楕円曲線とするような曲面である。 ファイバーが楕円曲線とならない点を特異ファイバー (singular fibers) と呼び、小平邦彦により分類された。弦理論の脈絡では、楕円ファイバーも特異ファイバーも (F-theory) を使う記述にとっても重要である。 楕円曲面は、曲面の興味深い例の多くを含む、曲面の大きなクラスで、複素幾何学の観点からも滑らかな(smooth) 4次元多様体の理論の観点からも、比較的良く理解されている。楕円曲面は代数体上の楕円曲線に似ている（つまり、類似している）。 (ja)
- ( 비슷한 이름의 타원면에 관해서는 해당 문서를 참조하십시오.) 대수기하학에서 타원 곡면(橢圓曲面, 영어: elliptic surface)은 거의 모든 곳에서 타원 곡선을 올로 하는 올다발이 주어진 곡면이다. (ko)
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