# Moscow Mathematical Journal

Volume 15, Issue 1, January–March 2015 pp. 31–48.

On Projections of Smooth and Nodal Plane Curves

**Authors**:
Yu. Burman (1) and Serge Lvovski (2)

**Author institution:**(1) National Research University Higher School of Economics, International Laboratory of Representation Theory and Mathematical Physics, 20 Myasnitskaya Ulitsa, Moscow 101000, Russia, and the Indepdendent University of Moscow, 11, B.Vlassievsky per., Moscow, Russia, 119002

(2) National Research University Higher School of Economics (HSE), AG Laboratory, HSE, 7 Vavilova str., Moscow, Russia, 117312

**Summary: **

Suppose that *C* ⊂ ℙ^{2} is a general enough nodal plane curve
of degree > 2, ν: *C*̂ → *C* is its normalization, and π: *C*′ → ℙ^{1} is a finite
morphism simply ramified over the same set of points as a projection
pr_{p}∘ν: *C*̂ → ℙ^{1}, where *p* ∈ ℙ^{2}\*C* (if deg *C* = 3, one should assume in
addition that deg π = 4). We prove that the morphism π is equivalent to
such a projection if and only if it extends to a finite morphism *X* → (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 ≥3 except for duals to smooth cubics; this strengthens one of Victor Kulikov's results.

2010 Math. Subj. Class. Primary: 14H50; Secondary: 14D05, 14N99.

**Keywords:**Plane algebraic curve, projection, monodromy, Picard–Lefschetz theory, Chisini conjecture.

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