环面簇中有理曲线的法丛分裂层
Normal Bundle Splitting Strata of Rational Curves in Toric Varieties
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中文总结 AI 辅助
本文研究环面簇中有理曲线法丛分裂型的分层，通过Cox构造和相对上同调态射给出非平衡性数值判据，并应用于射影空间爆破，与已有结果等价。
中文摘要 AI 辅助
我们研究了在代数闭域（任意特征）上的光滑射影环面簇中，固定类无分歧有理曲线族按法丛分裂型的分层。利用Cox构造，我们在固定源参数空间上的万有曲线上，通过线丛直和构造了相对法丛的显式两项消解。然后我们在参数空间上构造向量丛态射，称为相对上同调态射，由此可以研究族的通用分裂型和跳跃轨迹。特别地，我们获得了法丛非平衡性的充分数值判据。我们将此方法应用于射影空间沿线性子空间的爆破，并证明非平衡性的数值判据与Cela--Lian \cite{CelaLian2026}的判据等价。我们还获得了跳跃轨迹的预期Chow类的显式公式。
英文摘要
We study stratification by normal-bundle splitting type in families of unramified rational curves of fixed class in a smooth projective toric variety over an algebraically closed field {of arbitrary characteristic}. Using the Cox construction, we construct an explicit two-term resolution of the relative normal bundle by direct sums of line bundles on the universal curve over the fixed-source parameter space. Then we construct morphisms of vector bundles on the parameter space, which we call relative cohomology morphisms, from which we can investigate generic splitting type and jumping loci of the family. In particular, we obtain sufficient numerical criteria for unbalancedness of the normal bundle. We apply this method to blowups of projective space along linear subspaces, and show that the numerical criteria for unbalancedness are equivalent to those of Cela--Lian \cite{CelaLian2026}. We also obtain explicit formulas for the expected Chow classes of the jump loci.
