两条平面曲线补集的不变双喷管与有效双曲性
Invariant two-jets and effective hyperbolicity for complements of two plane curves
- Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
- University of Chinese Academy of Sciences（中国科学院大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对射影平面中两条光滑曲线补集，在特定次数条件下证明有效第二主定理，给出对应常数，结合陈希的轨迹得到补集的小林双曲性与双曲嵌入，通过构造不变双喷管微分等完成证明。
AI中文摘要：
设$D=C_1+C_2$为射影平面$\boldsymbol{P}^2$中光滑平面曲线的简单正常相交并，次数满足$1\leqslant d_1\leqslant d_2$。我们证明，当次数对满足$d_1,d_2\geqslant3$，或$d_1=2$且$d_2\geqslant5$，或$d_1=1$且$d_2\geqslant8$时，对一般有序对成立有效第二主定理。对每个容许次数对，存在非空扎里斯基开集的有序对$(C_1,C_2)$，使得每个代数非退化整曲线$f:\mathbb{C}\to\boldsymbol{P}^2$（其像不包含于$D=C_1+C_2$）满足$T_f(r)\leqslant \mathcal{A}_{d_1,d_2}N_f^{[1]}(r,D)+o(T_f(r))$。对两条三次曲线，可取$\boldsymbol{A}_{3,3}=57$；对一条二次曲线与一条五次曲线，$\boldsymbol{A}_{2,5}=45$；对一条直线与一条八次曲线，$\boldsymbol{A}_{1,8}=69$。将所得扎里斯基开参数轨迹与陈希的极一般代数双曲性轨迹相交，可得到补集的小林双曲性与双曲嵌入。证明过程首先构造一条负扭曲的不变双喷管微分，随后通过Demailly–El Goul零轨迹论证或混合$\boldsymbol{O}_{\boldsymbol{P}^2}(3)$斜向量场求导得到第二个方程，仅需对短列表的低扭曲进行有限计算，在这些情形下，有限域上的精确秩证书可证明所需的关键消失引理。
英文摘要：
Let $D=C_1+C_2\subset\mathbb{P}^2$ be a simple-normal-crossing union of smooth plane curves of degrees $1\leqslant d_1\leqslant d_2$. We prove an effective Second Main Theorem on a nonempty Zariski-open set of ordered pairs whenever \[ d_1,d_2\geqslant3, \qquad\text{or}\qquad d_1=2,\ d_2\geqslant5, \qquad\text{or}\qquad d_1=1,\ d_2\geqslant8. \] After intersecting with the standard very-general algebraic-hyperbolicity locus, we obtain Kobayashi hyperbolicity and hyperbolic embedding of the complement. The proof uses logarithmic invariant two-jet differentials, the corrected Demailly--El Goul zero-locus argument, and a two-component adaptation of slanted vector fields with base twist $\mathcal{O}_{\mathbb{P}^2}(3)$. The adapted twist three is uniform in $(d_1,d_2)$: simultaneous cancellation of the two root equations requires third Taylor differences, but no seventh-order polarization. For a linear component the third-remainder space vanishes, and ten compatible mixed generators replace the generic Taylor package. A numerical phase diagram shows that exact finite computation is needed only for the six boundary pairs $(3,3),(3,4),(2,6),(2,5),(1,9),(1,8)$. Their key vanishings are certified by full-column-rank matrices over finite fields. For two cubics, a strengthened range reduces to twelve theorem-relevant degree--twist cases; two low-weight self-tests bring the computational total to $168$ character--eigenvalue blocks. The exact source, certificates, and reproduction instructions form an accompanying computational archive. In the two-cubic case the resulting explicit estimate is \[ T_f(r)\leqslant57\bigl(N_f^{[1]}(r,C_1)+N_f^{[1]}(r,C_2)\bigr) +o(T_f(r))\ \|. \] For a conic and a quintic one may take the explicit coefficient $45$, and for a line and an octic one may take $69$.
