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Felix Klein's paper on real flexes vindicated

Felice Ronga (1998)

Banach Center Publications

In a paper written in 1876 [4], Felix Klein gave a formula relating the number of real flexes of a generic real plane projective curve to the number of real bitangents at non-real points and the degree, which shows in particular that the number of real flexes cannot exceed one third of the total number of flexes. We show that Klein's arguments can be made rigorous using a little of the theory of singularities of maps, justifying in particular his resort to explicit examples.

Field of moduli versus field of definition for cyclic covers of the projective line

Aristides Kontogeorgis (2009)

Journal de Théorie des Nombres de Bordeaux

We give a criterion, based on the automorphism group, for certain cyclic covers of the projective line to be defined over their field of moduli. An example of a cyclic cover of the complex projective line with field of moduli ℝ that can not be defined over ℝ is also given.

Fields of moduli of three-point G -covers with cyclic p -Sylow, II

Andrew Obus (2013)

Journal de Théorie des Nombres de Bordeaux

We continue the examination of the stable reduction and fields of moduli of G -Galois covers of the projective line over a complete discrete valuation field of mixed characteristic ( 0 , p ) , where G has a cyclic p -Sylow subgroup P of order p n . Suppose further that the normalizer of P acts on P via an involution. Under mild assumptions, if f : Y → ℙ 1 is a three-point G -Galois cover defined over ℚ ¯ , then the n th higher ramification groups above p for the upper numbering of the (Galois closure of the) extension K / ℚ vanish,...

Finite subschemes of abelian varieties and the Schottky problem

Martin G. Gulbrandsen, Martí Lahoz (2011)

Annales de l’institut Fourier

The Castelnuovo-Schottky theorem of Pareschi-Popa characterizes Jacobians, among indecomposable principally polarized abelian varieties ( A , Θ ) of dimension g , by the existence of g + 2 points Γ ⊂ A in special position with respect to 2 Θ , but general with respect to Θ , and furthermore states that such collections of points must be contained in an Abel-Jacobi curve. Building on the ideas in the original paper, we give here a self contained, scheme theoretic proof of the theorem, extending it to finite, possibly...

Finiteness results for Hilbert's irreducibility theorem

Peter Müller (2002)

Annales de l’institut Fourier

Let k be a number field, 𝒪 k its ring of integers, and f ( t , X ) ∈ k ( t ) [ X ] be an irreducible polynomial. Hilbert’s irreducibility theorem gives infinitely many integral specializations t ↦ t ¯ ∈ 𝒪 k such that f ( t ¯ , X ) is still irreducible. In this paper we study the set Red f ( 𝒪 k ) of those t ¯ ∈ 𝒪 k with f ( t ¯ , X ) reducible. We show that Red f ( 𝒪 k ) is a finite set under rather weak assumptions. In particular, previous results obtained by diophantine approximation techniques, appear as special cases of some of our results. Our method is different. We use elementary group...

Five-gonal curves of genus nine.

Marc Coppens (2005)

Collectanea Mathematica

Let C be a smooth 5-gonal curve of genus 9. Assume all linear systems g15 on C are of type I (i.e. they can be counted with multiplicity 1) and let m be the numer of linear systems g15 on C. The only possibilities are m=1; m=2; m=3 and m=6. Each of those possibilities occur.

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