发表机构
Università degli Studi di Milano; Universität Bonn（米兰大学; 波恩大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究K3曲面及超曲面超平面截面的模映射的分歧性，证明一般二次K3曲面的本原模映射在171个点处分歧，而一般四次K3曲面的模映射无分歧，并探讨高倍数极化的分歧与切丛稳定性的关联。
AI 中文摘要
与K3曲面上的线性系统相关的模态射$\u03bc_n\uff1a|nH|_{\rm{sm}}\to\mathcal{M}_{g_n}$的有限性和单射性在许多情形下已被理解。本文研究其分歧性。已有的消没和稳定性结果已表明，当极化或倍数足够正时，映射无分歧，因此我们关注低次现象。特别地，对于一般的二次K3曲面，我们证明本原模映射是拟有限的，但恰好在$171$个点处分歧，这些点被识别为与分支六次曲线相关的对数切丛的跳跃线。另一方面，我们证明若$X\subset\mathbb{P}^n$是一般次数至少为$4$的超曲面，则对$X$的每个光滑超平面截面$Y$，有$\rm{H}^0(Y,T_X|_Y)=0$。特别地，一般四次K3曲面的光滑超平面截面的模映射是无分歧的。此外，我们还表明，在二次和四次K3曲面上，对于本原极化的更高倍数，分歧确实会发生，并将该问题与切丛的稳定性联系起来。
英文摘要
Finiteness and injectivity of the moduli morphisms $μ_n\colon|nH|_{\rm{sm}}\to\mathcal{M}_{g_n}$ associated with linear systems on K3 surfaces are understood in many cases. In this paper we study their ramification. Existing vanishing and stability results already imply unramifiedness when the polarisation or the multiple is sufficiently positive, so our focus is on low-degree phenomena. In particular, for a general K3 surface of degree $2$, we prove that the primitive moduli map is quasi-finite but ramified at exactly $171$ points, identified with the jumping lines of a logarithmic tangent bundle associated with the branch sextic. On the other hand, we prove that if $X\subset\mathbb{P}^n$ is a general hypersurface of degree at least $4$, then $\rm{H}^0(Y,T_X|_Y)=0$ for every smooth hyperplane section $Y$ of $X$. In particular, the moduli map for smooth hyperplane sections of a general quartic K3 surface is unramified. We moreover show that ramification does occur for higher multiples of the primitive polarisation on both degree-$2$ and degree-$4$ K3 surfaces, and relate the question to the stability of the tangent bundle.
