On projections of smooth and nodal plane curves

Suppose that $C\subset\mathbb P^2$ is a general enough nodal plane curve of degree $>2$, $ν\colon \hat C\to C$ is its normalization, and $π\colon \hat C\to\mathbb P^1$ is a finite morphism simply ramified over the same set of points as a projection $\mathrm{pr}_p\circ ν\colon\hat C \to\mathbb P^1$, where $p\in\mathbb P^2\setminus C$ (if $\mathrm{deg}\, C=3$, one should assume in addition that $\degπ\ne4$). We prove that the morphism $π$ is equivalent to such a projection if and only if it extends to a finite morphism $X\to(\mathbb P^2)^*$ ramified over $C^*$, 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 $\ge3$ except for duals to smooth cubics; this strengthens one of Victor Kulikov's results.

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