摘要翻译：
设X是光滑射影4重Y沿光滑曲线C的爆破，设E是例外因子。假设X是一个Fano流形，并且具有（3,1）型的初等极值压缩$\phi:X\to Z$，使得E是$\phi$-充足的（回想一下，如果例外轨迹是一个除数，并且它的图像是一条曲线，则4倍的压缩映射称为（3,1）-型）。我们证明了如果$\phi$的例外因子是光滑的，则Y与$\mathbb{P}^{4}$同构，C是4次椭圆曲线。
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英文标题：
《A remark on Fano 4-folds having (3,1)-type extremal contractions》
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作者：
Toru Tsukioka
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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 X be the blow-up of a smooth projective 4-fold Y along a smooth curve C and let E be the exceptional divisor. Assume that X is a Fano manifold and has an elementary extremal contraction $\phi: X \to Z$ of (3,1)-type such that E is $\phi$-ample (recall that a contraction map for a 4-fold is called (3,1)-type if the exceptional locus is a divisor and its image is a curve). We show that if the exceptional divisor of $\phi$ is smooth, then Y is isomorphic to $\mathbb{P}^{4}$ and C is an elliptic curve of degree 4.
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PDF链接：
https://arxiv.org/pdf/0710.1719