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摘要翻译：
设W->X是由有理曲线构成的实光滑射影3倍纤维。J.Koll\'ar证明，如果W(R)是可定向的，则W(R)的连通分量N本质上是Seifert纤维流形或透镜空间的连通和。我们的主要定理在回答Koll\'ar的三个肯定问题时，给出了当X是几何有理曲面时Seifert纤维的个数和重数以及透镜空间的扭转数和扭转数的精确估计。当N在基轨道F上是Seifert纤维时，我们的结果推广了光滑实有理曲面上的Comessatti定理：F不能同时是可定向的和双曲型的。我们惊奇地表明，与Comessatti定理不同，有一些例子，其中F是不可定向的，是双曲型的，X是极小的。我们使用的技术是将Seifert纤维构造为Du Val曲面的投影切线束。
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英文标题：
《Real singular Del Pezzo surfaces and 3-folds fibred by rational curves,
II》
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作者：
Fabrizio Catanese and Frederic Mangolte
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最新提交年份：
2008
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分类信息：
一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Mathematics 数学
二级分类：Geometric Topology 几何拓扑
分类描述：Manifolds, orbifolds, polyhedra, cell complexes, foliations, geometric structures
流形，轨道，多面体，细胞复合体，叶状，几何结构
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英文摘要：
Let W -> X be a real smooth projective 3-fold fibred by rational curves. J. Koll\'ar proved that, if W(R) is orientable, then a connected component N of W(R) is essentially either a Seifert fibred manifold or a connected sum of lens spaces. Our Main Theorem, answering in the affirmative three questions of Koll\'ar, gives sharp estimates on the number and the multiplicities of the Seifert fibres and on the number and the torsions of the lens spaces when X is a geometrically rational surface. When N is Seifert fibred over a base orbifold F, our result generalizes Comessatti's theorem on smooth real rational surfaces: F cannot be simultaneously orientable and of hyperbolic type. We show as a surprise that, unlike in Comessatti's theorem, there are examples where F is non orientable, of hyperbolic type, and X is minimal. The technique we use is to construct Seifert fibrations as projectivized tangent bundles of Du Val surfaces.
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PDF链接：
https://arxiv.org/pdf/0803.2074
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