Given a parameterization $\phi$ of a rational plane curve C, we study some invariants of C via $\phi$. We first focus on the characterization of rational cuspidal curves, in particular we establish a relation between the discriminant of the pull-back of a line via $\phi$, the dual curve of C and its singular points. Then, by analyzing the pull-backs of the global differential forms via $\phi$, we prove that the (nearly) freeness of a rational curve can be tested by inspecting the Hilbert function of the kernel of a canonical map. As a by product, we also show that the global Tjurina number of a rational curve can be computed directly from one of its parameterization, without relying on the computation of an equation of C.

中文翻译：

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有理平面曲线的自由度和不变量

给定有理平面曲线 C 的参数化 $\phi$，我们通过 $\phi$ 研究 C 的一些不变量。我们首先关注有理尖牙曲线的表征，特别是我们通过$\phi$建立了一条线的回拉判别式、C的对偶曲线及其奇异点之间的关系。然后，通过通过 $\phi$ 分析全局微分形式的回拉，我们证明可以通过检查典型映射核的希尔伯特函数来测试有理曲线的（几乎）自由度。作为副产品，我们还表明，有理曲线的全局 Tjurina 数可以直接从其参数化之一计算出来，而无需依赖于 C 方程的计算。