On projections of smooth and nodal plane curves

TL;DR

This paper characterizes when certain finite morphisms from the normalization of a nodal plane curve to the projective line are equivalent to projections from points outside the curve, and proves the Chisini conjecture for a broad class of ramified mappings.

Contribution

It establishes a criterion for morphisms to be equivalent to projections based on their extension to a smooth surface, and proves the Chisini conjecture for duals of general nodal curves of degree at least three.

Findings

Morphisms extend to smooth surfaces iff equivalent to projections.

Chisini conjecture proven for duals of general nodal curves of degree ≥3.

Strengthens previous results by Victor Kulikov.

Abstract

Suppose that $C\subset\mathbb P^2$ is a general enough nodal plane curve of degree $>2$, $\nu\colon \hat C\to C$ 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 $\mathrm{pr}_p\circ \nu\colon\hat C \to\mathbb P^1$, where $p\in\mathbb P^2\setminus C$ (if $\mathrm{deg}\, C=3$, one should assume in addition that $\deg\pi\ne4$). We prove that the morphism $\pi$ is equivalent to such a projection if and only if it extends to a finite morphism $X\to(\mathbb P^2)^*$ 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.