AI 中文总结
研究半代数曲线配置的拓扑，通过局部节点贡献ψ发展组合理论，证明区域计数公式等，在代数设置中研究相关条件及差异，引入二项式节点不变量，揭示精确性障碍与关联数据关系及\(\psi\)与不变量关系。
AI 中文摘要
我们为具有普通多重交点的连通半代数曲线的有限配置发展了一种组合理论，由局部节点贡献ψ控制，它决定全局几何和拓扑性质。我们证明了精确的区域计数公式，刻画了最大配置，并将删除 - 限制递归扩展到一般曲线配置。在代数设置中，我们证明没有三重交点（\(k_x = 3\)）是OS型代数通过上同调分解的充分条件，反之不成立。对于线配置，我们计算了简化的OS模型与\(H^2\)之间的差异，表明它由\(k_x\geq4\)的节点控制，等于\(\sum_{k_x\geq4}\binom{k_x - 1}{2}\)。节点贡献通过欧拉特征出现在混合霍奇结构中，对于线配置，等于\(\dim \operatorname{Gr}_4^W H^2\)。并发线的缺陷复形表明精确性障碍需要曲线方向的关联数据。最后，我们引入二项式节点不变量\(\{\Psi_k\}\)，证明\(\Psi_2\)是通用的线性局部加性不变量，并表明\(\psi=\Psi_1 - \Psi_0\)。
英文摘要
We develop a combinatorial theory for finite arrangements of connected semialgebraic curves with ordinary multiple intersections, governed by a local node contribution $ψ$ that determines global geometric and topological properties. We prove exact region-count formulas, characterize maximal arrangements, and extend the deletion--restriction recurrence to general curve arrangements. In the algebraic setting, we prove that the absence of triple points ($k_x=3$) is a sufficient condition for the OS-type algebra to factor through cohomology; the converse, however, fails already for line arrangements, where the classical Orlik--Solomon relations ensure factorization even in the presence of triple points. For line arrangements we compute the discrepancy between the simplified OS-model and $H^2$, showing it is governed by nodes with $k_x\ge4$ and equals $\sum_{k_x\ge4}\binom{k_x-1}{2}$. The node contribution appears in the mixed Hodge structure via the Euler characteristic; for arrangements in normal crossing position we compute the full weight decomposition of $H^2$ and show it is Hodge--Tate exactly when every component has genus zero, recovering the line-arrangement case as $\dim\operatorname{Gr}^W_4H^2=ψ$. The defect complex for concurrent lines reveals that exactness obstructions require curve-wise incidence data. Finally, we introduce binomial node invariants $\{Ψ_k\}$, prove $Ψ_2$ is the universal linearly locally additive invariant, and show $ψ=Ψ_1-Ψ_0$.
