非真概型上的丰富向量丛
Ample vector bundles on non-proper schemes
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中文总结 AI 辅助
该研究解决Hartshorne提出的两个公开问题，证明任意特征下丰富向量丛张量积的丰富性，还构造出光滑拟射影曲面中丰富线丛的非丰富自扩张。
中文摘要 AI 辅助
我们解决了Hartshorne在1966年提出的关于丰富向量丛的两个公开问题。我们证明了在代数闭域上有限型概型上，丰富向量丛的张量积在任意特征下都是丰富的，这推广了Hartshorne的特征零结果和Barton的射影正特征定理。更一般地，设f: X → S为概型的态射，我们证明f-丰富向量丛的张量积是f-丰富的。此外，若E是秩r>0的f-丰富向量丛，且W是正秩的有限局部自由多项式GL(r,S)-模且W₀=0，则E(W)是f-丰富的；特别地，对任意n>0，对称幂Gammaⁿ E是f-丰富的。最后，改编Ejiri-Fujino-Iwai的构造，我们证明在任意特征下，光滑拟射影曲面都存在一个由丰富线丛自身扩张得到的非丰富扩张。
英文摘要
We solve two open problems on ample vector bundles posed by Hartshorne in 1966. We prove that tensor products of ample vector bundles on schemes of finite type over an algebraically closed field are ample in arbitrary characteristic, extending Hartshorne's characteristic-zero result and Barton's projective positive-characteristic theorem. More generally, let f: X -> S be a morphism of schemes. Then we prove that tensor products of f-ample vector bundles are f-ample. Moreover, if E is an f-ample vector bundle of rank r>0 and W is a finite locally free polynomial GL(r,S)-module of positive rank with W_0 = 0, then E(W) is f-ample. In particular, Gamma^n E is f-ample for every n>0. Finally, adapting a construction of Ejiri-Fujino-Iwai, we show that in every characteristic a smooth quasi-projective surface carries a non-ample extension of an ample line bundle by itself.
发表机构
- University of Warsaw（华沙大学）
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