Existence of Rational Curves on Algebraic Varieties, Minimal Rational Tangents, and Applications
Existence of Rational Curves on Algebraic Varieties, Minimal Rational Tangents, and Applications
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代数簇有理曲线的存在性、最小有理切线及其应用
DOI：
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发表时间：
2005
期刊：
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通讯作者：
L. S. Conde
中科院分区：
文献类型：
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作者：
Stefan Kebekus;L. S. Conde
本文综述了代数簇上有理曲线研究的一些最新进展。它是为优先方案“复杂几何的全球方法”的调查卷编写的，得到DFG的支持。
首先，我们讨论的存在性结果的有理曲线的叶状流形。我们构建的切线态射，定义各种最小有理切线（VMRT）和名称的一些结果表明，许多的几何性质的uniruled品种编码在射影几何的VMRT。
然后将结果应用于两种不同的设置。首先，我们讨论了uniruled品种上的有理曲线链的几何，展示了如何从VMRT确定品种的长度，并讨论了曲线上向量丛的模空间的具体例子。
另一方面，有理曲线的存在性结果也可以用来研究已知不含有理曲线的流形。我们利用这种方法来研究态射的变形。最后，我们期待在家庭的品种的正则极化流形在复杂的表面，并使用不存在的结果，合理的曲线的基础上，涉及家庭的变化与对数科代拉维的基础。
This survey paper discusses some of the recent progress in the study of rational curves on algebraic varieties. It was written for the survey volume of the priority programme "Global Methods in Complex Geometry", supported by the DFG.
To start, we discuss existence results for rational curves on foliated manifolds. We construct the tangent morphism, define the variety of minimal rational tangents (VMRT) and name a number of results that show that many of the geometry properies of a uniruled variety are encoded in the projective geometry of the VMRT.
The results are then applied in two different settings. For one, we discuss the geometry of chains of rational curves on uniruled varieties, show how the length of a variety can be determined from the VMRT and discuss the concrete example of the moduli space of vector bundles on a curve.
On the other hand, the existence results for rational curves can be also be used to study manifolds which are known NOT to contain rational curves. We employ this method to study deformations of morphisms. Finally, we look at families of varieties of canonically polarized manifolds over complex surfaces, and use non-existence results for rational curves on the base to relate the variation of the family with the logarithmic Kodaira dimension of the base.