On projections of smooth and nodal plane curves

Suppose that $C\subset {\mathbb{P}}^2$ is a general enough nodal plane curve of degree $>2$, $\nu \text{\hspace{0.17em}⁣}:\widehat{C}\to C$ is its normalization, and $\pi\colon \hat C\to\mathbb P^1$ is a finite morphism simply ramified over the same set of points as a projection ${\mathrm{p}\mathrm{r}}_p\circ \nu \text{\hspace{0.17em}⁣}:\widehat{C}\to {\mathbb{P}}^1$, where $p\in {\mathbb{P}}^2\setminus C$ (if $\mathrm{d}\mathrm{e}\mathrm{g}C=3$, one should assume in addition that $\mathrm{deg}\pi \mathrm{\ne }4$). We prove that the morphism $\pi$ is equivalent to such a projection if and only if it extends to a finite morphism $X\to ({\mathbb{P}}^2{)}^{\ast }$ ramified over $C^{\ast }$, where $X$ is a smooth surface. As a by-product, we prove the Chisini conjecture for mappings ramified over duals to general nodal curves of any degree $\ge 3$ except for duals to smooth cubics; this strengthens one of Victor Kulikov’s results.