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 Mosc. Math. J., 2015, Volume 15, Number 1, Pages 31–48 (Mi mmj547)

On projections of smooth and nodal plane curves

Yu. Burmanab, Serge Lvovskic

a Indepdendent University of Moscow, 11, B. Vlassievsky per., Moscow, Russia, 119002
b National Research University Higher School of Economics, International Laboratory of Representation Theory and Mathematical Physics, 20 Myasnitskaya Ulitsa, Moscow 101000, Russia
c National Research University Higher School of Economics (HSE), AG Laboratory, HSE, 7 Vavilova str., Moscow, Russia, 117312

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 $\pi\colon C'\to\mathbb P^1$ 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 $\deg C=3$, 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.

Key words and phrases: plane algebraic curve, projection, monodromy, Picard–Lefschetz theory, Chisini conjecture.

DOI: https://doi.org/10.17323/1609-4514-2015-15-1-31-48

Full text: http://www.mathjournals.org/.../2015-015-001-002.html
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MSC: Primary 14H50; Secondary 14D05, 14N99
Received: April 16, 2014; in revised form October 16, 2014
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Citation: Yu. Burman, Serge Lvovski, “On projections of smooth and nodal plane curves”, Mosc. Math. J., 15:1 (2015), 31–48

Citation in format AMSBIB
\Bibitem{BurLvo15} \by Yu.~Burman, Serge~Lvovski \paper On projections of smooth and nodal plane curves \jour Mosc. Math.~J. \yr 2015 \vol 15 \issue 1 \pages 31--48 \mathnet{http://mi.mathnet.ru/mmj547} \crossref{https://doi.org/10.17323/1609-4514-2015-15-1-31-48} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3427410} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000354886200002} 

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This publication is cited in the following articles:
1. Yu. Burman, B. Shapiro, “On Hurwitz-Severi numbers”, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 19:1 (2019), 155–167