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Michael Lönne

Publications and source records attributed to Michael Lönne.

At least 19 recordsLinked to original sources
arXiv2025

A faithful action of Gal($\overline{\mathbb{Q}}/\mathbb{Q}$) on Zariski multiplets

In this work, we establish two main results in the context of arithmetic and geometric properties of plane curves. First, we construct numerous new examples of arithmetic Zariski pairs and multiplets, where only a few ones were previously available. Second, we describe a faithful action of the absolute Galois group on the equisingular strata of plane curves, providing insights into the interplay between Galois representations and the geometry of singular plane curves. We conclude the paper with very concretes examples of the general results obtained.

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arXiv2025

$π_1$ of trigonal loci of strata of abelian differentials

We investigate locally closed subspaces of projectivized strata of abelian differentials which classify trigonal curves with canonical divisor a multiple of a trigonal divisor. We describe their orbifold structure using linear systems on Segre-Hirzebruch surfaces and obtain results for their orbifold fundamental groups. Most notable among these orbifolds is the connected component $\mathbf P\mathcal H^{ev}_4(6)$, the projectivisation of the space $\mathcal H^{ev}_4(6)$ of abelian differentials on non-hyperelliptic genus $4$ curves with a single zero of multiplicity 6 providing an even spin structure. Its orbifold fundamental group is identified with the quotient of the Artin group of type $E_8$ by its maximal central subgroup.

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arXiv2023

On Elliptic K3 Surfaces and Dessins d'Enfants

We classify subgroups of $\textrm{SL}(2,\mathbb{Z})$ up to conjugacy, which occur as monodromy groups of elliptically fibered K3 surfaces following a general strategy proposed by Bogomolov and Tschinkel. The essential step is the factorisation of the functional invariant $j$ with second factor a Belyi function of maximal possible degree and the classification of the corresponding subgroups $\barΓ$ of $\textrm{PSL}(2,\mathbb{Z})$ using dessins d'enfants.

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arXiv2021

Secondary Braid Groups

We generalize presentations of the fundamental group of discriminant complements and arrive at a class of presentations associated naturally with words in the free monoid of the alphabet $σ_1,\dots,σ_{n-1}$. Our study addresses invariance properties of these presentations and the presented groups under various operations on the words. In particular we prove that the group does only depend on the corresponding element in the positive braid monoid, and under mild hypotheses only on its conjugacy class in the braid group.

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arXiv2021

Moduli of elliptic $K3$ surfaces: monodromy and Shimada root lattice strata

In this paper we investigate two stratifications of the moduli space of elliptically fibred K3 surfaces. The first comes from Shimada's classification of connected components of elliptically fibred K3 surfaces and is closely related to the root lattice of the fibration. The second is the monodromy stratification defined by Bogomolov, Petrov and Tschinkel. The main result of the paper is a classification of all positive-dimensional ambi-typical strata, that is strata which are both Shimada root strata and monodromy strata. We further discuss the connection with moduli spaces of lattice-polarised K3 surfaces. The paper contains an appendix by M. Kirschmer providing computational results on the 1-dimensional ambi-typical strata.

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arXiv2019

Branch Stabilisation for the Components of Hurwitz Moduli Spaces of Galois Covers

We consider components of Hurwitz moduli space of G-Galois covers and set up a powerful algebraic framework to study the set of corresponding equivalence classes of monodromy maps. Within that we study geometric stabilisation by various G-covers branched over the disc. Our results addresses the problem to decide equivalence and stable equivalence algebraically. We recover a homological invariant, which we show to distinguish the equivalence classes of given boundary monodromy and Nielsen type, if the latter is sufficiently large in the appropriate sense.

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arXiv2018

On Zariski Multiplets of Branch Curves from Surfaces Isogenous to a Product

In this paper we give an asymptotic bound of the cardinality of Zariski multiples of particular plane singular curves. These curves have only nodes and cusps as singularities and are obtained as branched curves of ramified covering of the plane by surfaces isogenous to a product of curves with group $(\mathbb{Z}/2\mathbb{Z})^k$. The knowledge of the moduli space of these surfaces will enable us to produce Zariski multiplets whose number grows subexponentialy in function of their degree.

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arXiv2018

On a discriminant knot group problem of Brieskorn

Quite some time ago, at the singularity conference at Cargèse 1972 Brieskorn asked the following question: Is the local fundamental group $π_1^{s}(S-D)$ of the discriminant complement inside the semi-universal unfolding $S$ of an isolated hypersurface singularity constant for $s$ in the $μ$-constant stratum $Σ_E$? We review this question and give an affirmative answer in case of singular plane curve germs of multiplicity at most $3$.

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arXiv2017

Double Kodaira fibrations with small signature

Kodaira fibrations are surfaces of general type with a non-isotrivial fibration, which are differentiable fibre bundles. They are known to have positive signature divisible by $4$. Examples are known only with signature 16 and more. We review approaches to construct examples of low signature which admit two independent fibrations. Special attention is paid to ramified covers of product of curves which we analyse by studying the monodromy action for bundles of punctured curves. As a by-product we obtain a classification of all fix-point-free automorphisms on curves of genus at most $9$.

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arXiv2017

Topological equivalence of holomorphic foliation germs of rank $1$ with isolated singularity in the Poincaré domain

We show that the topological equivalence class of holomorphic foliation germs with an isolated singularity of Poincaré type is determined by the topological equivalence class of the real intersection foliation of the (suitably normalized) foliation germ with a sphere centered in the singularity. We use this Reconstruction Theorem to completely classify topological equivalence classes of plane holomorphic foliation germs of Poincaré type and discuss a conjecture on the classification in dimension $\geq 3$.

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arXiv2015

Mapping Class Groups of Trigonal Loci

In this paper we study the topology of the stack $\mathcal{T}_g$ of smooth trigonal curves of genus g, over the complex field. We make use of a construction by the first named author and Vistoli, that describes $\mathcal{T}_g$ as a quotient stack of the complement of the discriminant. This allows us to use techniques developed by the second named author to give presentations of the orbifold fundamental group of $\mathcal{T}_g$, of its substrata with prescribed Maroni invariant and describe their relation with the mapping class group $\mathcal{M}ap_g$ of Riemann surfaces of genus g.

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arXiv2013

Genus stabilization for moduli of curves with symmetries

In a previous paper, arXiv:1206.5498, we introduced a new homological invariant $\e$ for the faithful action of a finite group G on an algebraic curve. We show here that the moduli space of curves admitting a faithful action of a finite group G with a fixed homological invariant $\e$, if the genus g' of the quotient curve is sufficiently large, is irreducible (and non empty iff the class satisfies the condition which we define as 'admissibility'). In the unramified case, a similar result had been proven by Dunfield and Thurston using the classical invariant in the second homology group of G, H_2(G, \ZZ). We achieve our result showing that the stable classes are in bijection with the set of admissible classes $\e$.

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arXiv2011

Irreducibility of the space of dihedral covers of the projective line of a given numerical type

We show in this paper that the set of irreducible components of the family of Galois coverings of P^1_C with Galois group isomorphic to D_n is in bijection with the set of possible numerical types. In this special case the numerical type is the equivalence class (for automorphisms of D_n) of the function which to each conjugacy class \mathcal{C} in D_n associates the number of branch points whose local monodromy lies in the class \mathcal{C}.

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arXiv2010

Bifurcation braid monodromy of plane curves

We consider spaces of plane curves in the setting of algebraic geometry and of singularity theory. On one hand there are the complete linear systems, on the other we consider unfolding spaces of bivariate polynomials of Brieskorn-Pham type. For suitable open subspaces we can define the bifurcation braid monodromy taking values in the Zariski resp. Artin braid group. In both cases we give the generators of the image. These results are compared with the corresponding geometric monodromy. It takes values in the mapping class group of braided surfaces. Our final result gives a precise statement about the interdependence of the two monodromy maps. Our study concludes with some implication with regard to the unfaithfulness of the geometric monodromy and the - yet unexploited - knotted geometric monodromy, which takes the ambient space into account.

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arXiv2010

Moduli spaces of surfaces and monodromy invariants

This paper is a survey of the authors' recent results on "abc-surfaces" and the monodromy of their natural Lefschetz fibrations and projections to P^1 x P^1, see (arXiv:0910.2142). The results being surveyed explore various fundamental questions about complex deformation vs. symplectomorphism vs. diffeomorphism from the perspective of one explicit family of examples; the paper also contains some remarks which were not in earlier papers.

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arXiv2009

Moduli spaces and braid monodromy types of bidouble covers of the quadric

Bidouble covers $π: S \mapsto Q$ of the quadric Q are parametrized by connected families depending on four positive integers a,b,c,d. In the special case where b=d we call them abc-surfaces. Such a Galois covering $π$ admits a small perturbation yielding a general 4-tuple covering of Q with branch curve $\De$, and a natural Lefschetz fibration obtained from a small perturbation of the composition of $ π$ with the first projection. We prove a more general result implying that the braid monodromy factorization corresponding to $\De$ determines the three integers a,b,c in the case of abc-surfaces. We introduce a new method in order to distinguish factorizations which are not stably equivalent. This result is in sharp contrast with a previous result of the first and third author, showing that the mapping class group factorizations corresponding to the respective natural Lefschetz pencils are equivalent for abc-surfaces with the same values of a+c, b. This result hints at the possibility that abc-surfaces with fixed values of a+c, b, although diffeomorphic but not deformation equivalent, might be not canonically symplectomorphic.

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