AI 中文总结
本文通过三种互补构造研究正特征中的有限决定性与模态，建立了矩阵芽决定性与余维数的等价，并利用 Frobenius 代数与临界覆盖证明真模态与右模态一致，给出决定性界与有限参数化。
AI 中文摘要
我们通过三种互补的构造来研究正特征中的有限决定性与模态：给定射流的横截性、Frobenius 形变方向以及有限平坦临界覆盖。对于任意域上的矩阵芽，给定射流到秩分层上的横截性表明，有限左-右决定性与切像的有限余维数等价。类似的结果对于纯右等价、无限域上的单侧作用以及有限域上的指定范围内均成立。这些方法还给出了正维理想有限接触决定性的刻画，并推导出决定理想在期望高度、Cohen-Macaulay 性、约化性以及正规性方面的结果。对于特征 $p>0$ 的代数闭域上的孤立超曲面奇点，我们引入了一个 Frobenius 代数，其长度与 Loewy 结构控制着 Milnor 数、决定性与本质余秩。利用有限平坦临界覆盖，我们证明了真模态与右模态一致。由此可知，在固定特征中，有界右模态给出了与环境维数无关的决定性界，并且，在非奇异二次悬挂的意义下，给出了由模族构成的有限代数参数化。同样的 Frobenius 方法还给出了 $F$-跳跃数的统一有限性结果，并表明 $F$-纯阈值在单个 Frobenius 层次上即可确定。
英文摘要
We study finite determinacy and modality in positive characteristic through three complementary constructions: prescribed-jet transversality, Frobenius deformation directions, and a finite flat critical cover. For matrix germs over an arbitrary field, prescribed-jet transversality to the rank stratification shows that finite left-right determinacy is equivalent to finite codimension of the tangent image. Analogous results hold for pure right equivalence, for one-sided actions over infinite fields, and in specified ranges over finite fields. These methods also yield a characterization of finite contact determinacy for positive-dimensional ideals and imply expected-height, Cohen-Macaulayness, reducedness, and normality results for determinantal ideals. For isolated hypersurface singularities over an algebraically closed field of characteristic $p>0$, we introduce a Frobenius algebra whose length and Loewy structure govern the Milnor number, determinacy, and essential corank. Using a finite flat critical cover, we prove that proper modality coincides with right modality. It follows that, in fixed characteristic, bounded right modality gives an ambient-dimension-independent bound on determinacy and, up to nonsingular quadratic suspension, a finite algebraic parametrization by modular families. The same Frobenius methods yield uniform finiteness results for $F$-jumping numbers and show that the $F$-pure threshold is determined at a single Frobenius level.
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