Monodromy of projections of hypersurfaces

Maria Gioia Cifani,Riccardo Moschetti,Alice Cuzzucoli

doi:10.1007/s10231-021-01132-3

Maria Gioia Cifani, Riccardo Moschetti + Show 1 more

Open Access

https://doi.org/10.1007/s10231-021-01132-3

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Abstract

Let X be an irreducible, reduced complex projective hypersurface of degree d. A point P not contained in X is called uniform if the monodromy group of the projection of X from P is isomorphic to the symmetric group S_d. We prove that the locus of non-uniform points is finite when X is smooth or a general projection of a smooth variety. In general, it is contained in a finite union of linear spaces of codimension at least 2, except possibly for a special class of hypersurfaces with singular locus linear in codimension 1. Moreover, we generalise a result of Fukasawa and Takahashi on the finiteness of Galois points.

Highlights

This paper aims at studying the monodromy group of projections of irreducible and reduced complex projective hypersurfaces
This paper is motivated by the work of Pirola and Schlesinger [25], in which the authors consider the monodromy group of projections of any irreducible reduced
The case of curves has been solved by Pirola and Schlesinger: in the work [25], the authors proved that the locus of non-uniform points associated with projections of an irreducible reduced plane curve is finite

Summary

Introduction

This paper aims at studying the monodromy group of projections of irreducible and reduced complex projective hypersurfaces. The case of curves has been solved by Pirola and Schlesinger: in the work [25], the authors proved that the locus of non-uniform points associated with projections of an irreducible reduced plane curve is finite. The main result of this paper describes a property of the locus of non-uniform points for projective hypersurfaces: Theorem 1.2 Let X be an irreducible, reduced hypersurface of Pn+1 , n ≥ 2. As a consequence of the main result, we give in Proposition 4.8 a bound on the dimension of the locus of non-uniform points for smooth varieties; in particular, such locus is finite for smooth hypersurfaces. 4 is devoted to the proof of Theorem 1.2 about non-uniform points, followed by the consequences concerning general projections of smooth varieties. We will use integral instead of the equivalent notion of irreducible and reduced

Monodromy

Projections

Families of tangent lines

Branch locus

Focal loci of a family of lines

The locus of non‐uniform points