On projections of smooth and nodal plane curves

2013/11/08 by Yu. Burman, Burman, Yu., Serge Lvovski +1
Mathematics · #14N99 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Primary 14H50 #math.AG #msc:14D05 #msc:14H50 #msc:14N99 #secondary 14D05

paper · pdf · doi:10.48550/arxiv.1311.1904

Proofs simplified, a missing case supplied, and an important reference added

openalex publication_date 2013/11/08 · arxiv created 2014/01/21 · arxiv updated 2014/01/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

Suppose that C⊂\mathbb P² is a general enough nodal plane curve of degree >2, ν\colon C→ C is its normalization, and π\colon C→\mathbb P¹ is a finite morphism simply ramified over the same set of points as a projection pr_p∘ ν\colon C →\mathbb P¹, where p∈\mathbb P²∖ C (if deg C=3, one should assume in addition that degπ≠4). We prove that the morphism π is equivalent to such a projection if and only if it extends to a finite morphism X→(\mathbb P²)^* 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.

Related

• On Chisini's Conjecture. II
• On the image of a curve in a normal surface by a plane projection
• On ramified covers of the projective plane I: Segre's theory and classification in small degrees, with Appendix by Eugenii Shustin
• Chisini's conjecture for curves with singularities of type xⁿ=y^m
• On a Chisini Conjecture