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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三重纤维。Koll\'ar证明了如果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。设k:=k(N)为定义如下的整数：如果g:n->F是Seifert纤维，则定义k:=k(N)为g的多个纤维的个数；如果N是透镜空间的连通和，则定义k为与P^3(R)不同的透镜空间的个数。我们的主要定理说：如果X是几何有理曲面，那么k<=4。此外，我们还证明了如果F是S^1×S^1的差胚，则W(R)是连通的，且k=0。这些结果回答了Koll\'ar的两个肯定问题，他在1999年证明了k<=6，并提出4是锐界。我们从对只有Du Val奇点的实奇异Del Pezzo曲面的仔细研究中导出了该定理。
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英文标题：
《Real singular Del Pezzo surfaces and threefolds fibred by rational
curves, I》
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作者：
Fabrizio Catanese, Fr\'ed\'eric Mangolte
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最新提交年份：
2007
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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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英文摘要：
Let W -> X be a real smooth projective threefold fibred by rational curves. Koll\'ar proved that if W(R) is orientable a connected component N of W(R) is essentially either a Seifert fibred manifold or a connected sum of lens spaces. Let k : = k(N) be the integer defined as follows: If g : N -> F is a Seifert fibration, one defines k : = k(N) as the number of multiple fibres of g, while, if N is a connected sum of lens spaces, k is defined as the number of lens spaces different from P^3(R). Our Main Theorem says: If X is a geometrically rational surface, then k <= 4. Moreover we show that if F is diffeomorphic to S^1xS^1, then W(R) is connected and k = 0. These results answer in the affirmative two questions of Koll\'ar who proved in 1999 that k <= 6 and suggested that 4 would be the sharp bound. We derive the Theorem from a careful study of real singular Del Pezzo surfaces with only Du Val singularities.
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PDF链接：
https://arxiv.org/pdf/0705.0814
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