AI 中文总结
该代数几何研究针对整拟投影子概形，界定了一致型主部射流正维数零概形的概率，解决了Poonen算术Bertini猜想，还给出多超曲面雅可比秩退化轨迹概率界，证明采用过滤$Q$-进分解等技术。
AI 中文摘要
设$\u0007cX\u0007subseteq\u0007Pj^n_{\u0007Z}$为固定的整拟投影子概形，在$\u0007Z$上光滑且相对维数为$r$。对每个固定的$m\u0007ge1$，我们界定了次数为$d$的一致型在$\u0007cX_p$上的限制的第$m$阶主部射流具有正维数零概形的概率。该界为$C(d+1)^{N_m}p^{-λ_m(d)}$，其中$N_m=\u0007binom{r+m}{m}$，$λ_m(d)=\u0007floor{m(d+1)/(m+1)}$。当$m=1$时，这给出Bertini奇点轨迹估计$C(d+1)^{r+1}p^{-\u0007ceil{d/2}}$。它解决了Poonen的算术Bertini猜想5.2，且在提高次数阈值后，对每个固定的$A>0$都能得到$p^{-A}$的界。对于$c\u0007le r$个独立超曲面，正维数雅可比秩退化轨迹的概率同时被$C\u0007sum_i(d_i+1)^{r+1}p^{-\u0007ceil{d_i/2}}$和$C'(d_{\u0007min}+1)^{r+1}p^{-\u0007ceil{d_{\u0007min}/2}}$界定。证明使用了过滤$Q$-进分解、三角正规泰勒块以及复杂度一致可控的雅可比主元图。
英文摘要
Let $\cX\subseteq\Pj^n_{\Z}$ be a fixed integral quasiprojective subscheme, smooth over $\Z$ of relative dimension $r$. For each fixed $m\ge1$, we bound the probability that the $m$th principal-parts jet of the restriction of a uniform degree-$d$ form to $\cX_p$ has a positive-dimensional zero scheme. The bound is $C(d+1)^{N_m}p^{-λ_m(d)}$, where $N_m=\binom{r+m}{m}$ and $λ_m(d)=\floor{m(d+1)/(m+1)}$. For $m=1$, this gives the Bertini singular-locus estimate $C(d+1)^{r+1}p^{-\ceil{d/2}}$. It settles Poonen's arithmetic Bertini Conjecture~5.2 and, after increasing the degree threshold, yields $p^{-A}$ for every fixed $A>0$. For $c\le r$ independent hypersurfaces, the probability of a positive-dimensional Jacobian rank-degeneracy locus is bounded both by $C\sum_i(d_i+1)^{r+1}p^{-\ceil{d_i/2}}$ and by $C'(d_{\min}+1)^{r+1}p^{-\ceil{d_{\min}/2}}$. The proof uses filtered $Q$-adic decompositions, triangular normal Taylor blocks, and Jacobian-pivot charts of uniformly controlled complexity.
