arXiv research

Study trisections on rational elliptic surfaces to find new Zariski pairs.

Two unique conic-line arrangements with degree 9 are found.

In this paper, complement-equivalent arithmetic Zariski pairs will be exhibited answering in the negative a question by Eyral-Oka on these curves and their groups. A complement-equivalent arithmetic Zariski pair is a pair of complex projective plane curves having Galois-conjugate equations in some number field whose co…

Study conic line arrangements of degree 7, finding their topology and connected components.

Paper finds infinite pairs of fiber-type curves with same topology but different embeddings.

New invariant detects non-homeomorphic arrangements with similar coefficients.

Generic Hitchin representations generate dense subgroups.

New invariant identifies complex line arrangements with same combinatorics but different embeddings.

The invariant $\mathcal{I}(\mathcal{A},ξ,γ)$ was first introduced by E. Artal, V. Florens and the author. Inspired by the idea of G. Rybnikov, we obtain a multiplicativity theorem of this invariant under the gluing of two arrangements along a triangle. An application of this theorem is to prove that the extended Rybnik…

The paper explores conic-line arrangements via Poncelet's theorem and finds families of reducible curves.

We construct a topological invariant of algebraic plane curves, which is in some sense an adaptation of the linking number of knot theory. This invariant is shown to be a generalization of the I-invariant of line arrangements developed by the first author with Artal and Florens. We give two practical tools for computin…

Using the invariant developed in [6], we differentiate four arrangements with the same combinatorial information but in different deformation classes. From these arrangements, we construct four other arrangements such that there is no orientation-preserving homeomorphism between them. Furthermore, some couples of arran…

A central question in the study of line arrangements in the complex projective plane $\mathbb{CP}^2$ is: when does the combinatorial data of the arrangement determine its topological properties? In the present work, we introduce a topological invariant of complexified real line arrangements, the chamber weight. This in…

New representations of hyperbolic 3-manifold groups into larger groups.

In this work, we study a family of Cremona transformations of weighted projective planes which generalize the standard Cremona transformation of the projective plane. Starting from special plane projective curves we construct families of curves in weighted projective planes with special properties. We explain how to co…

A $k$-Artal arrangement is a reducible algebraic curve composed of a smooth cubic and $k$ inflectional tangents. By studying the topological properties of their subarrangements, we prove that for $k=3,4,5,6$, there exist Zariski pairs of $k$-Artal arrangements. These Zariki pairs can be distinguished in a geometric way…

We introduce an algorithm that exploits a combinatorial symmetry of an arrangement in order to produce a geometric reflection between two disconnected components of its moduli space. We apply this method to disqualify three real examples found in previous work by the authors from being Zariski pairs. Robustness is show…

We consider a certain hybridization construction which produces a subgroup of ${\rm PU}(n,1)$ from a pair of lattices in ${\rm PU}(n-1,1)$. Among the Picard modular groups ${\rm PU}(2,1,\mathcal{O}_d)$, we show that the hybrid of pairs of Fuchsian subgroups ${\rm PU}(1,1,\mathcal{O}_d)$ is a lattice when $d=1$ and $d=7…

Classifies Zariski closures of positive representations in Lie groups.

Study conic-line arrangements of degree 7, finding their topology and components.

The study finds conditions for certain groups to be dense in a specific mathematical space.

We show that in any $\mathbb{Q}$-Gorenstein flat family of klt singularities, normalized volumes are lower semicontinuous with respect to the Zariski topology. A quick consequence is that smooth points have the largest normalized volume among all klt singularities. Using an alternative characterization of K-semistabili…

New lattices in higher dimensions have dense surface subgroups.

Two arrangements with the same combinatorial intersection lattice but whose complements have different fundamental groups are called a Zariski pair. This work finds that there are at most nine such pairs amongst all ten line arrangements whose intersection points are doubles or triples. This result is obtained by consi…

We show that a surface group contained in a reductive real algebraic group can be deformed to become Zariski dense, unless its Zariski closure acts transitively on a Hermitian symmetric space of tube type. This is a kind of converse to a rigidity result of Burger, Iozzi and Wienhard.

Deforms surface groups to be Zariski dense in SL(n,R)

We prove that every Bers slice of quasi-Fuchsian space is Zariski dense in the character variety.

In a previous work, the third named author found a combinatorics of line arrangements whose realizations live in the cyclotomic group of the fifth roots of unity and such that their non-complex-conjugate embedding are not topologically equivalent in the sense that they are not embedded in the same way in the complex pr…

The paper finds dense subgroups in certain Lie groups.

We consider the Alexander polynomial of a plane algebraic curve twisted by a linear representation. We show that it divides the product of the polynomials of the singularity links, for unitary representations. Moreover, their quotient is given by the determinant of its Blanchfield intersection form. Specializing in the…

Let $S$ be an elliptic surface over a smooth curve $C$ with a section $O$. We denote its generic fiber by $E_S$. For a divisor $D$ on $S$, we canonically associate a $C(C)$-rational point $P_D$. In this note, we give a description of $P_D$ of $E_S$, when the rank of the group of $C(C)$-rational points is one. We apply …

Paper finds surface groups can deform in reductive symmetric spaces.

The paper finds free semigroups in dense subgroups of Lie groups with critical exponents arbitrarily close to the subgroup's.

We show that uniform K-stability is a Zariski open condition in Q-Gorenstein families of Q-Fano varieties. To prove this result, we consider the behavior of the stability threshold in families. The stability threshold (also known as the delta-invariant) is a recently introduced invariant that is known to detect the K-s…

We begin by showing that commensurators of Zariski dense subgroups of isometry groups of symmetric spaces of non-compact type are discrete provided that the limit set on the Furstenberg boundary is not invariant under the action of a (virtual) simple factor. In particular for rank one or simple Lie groups, Zariski dens…

Bi-Lipschitz rigidity theorem for dense subgroups of algebraic groups.

The paper connects arithmetic invariants of hyperbolic 3-manifolds.

Maximal representations in symplectic lattices proven for most cases.

We discuss properties of complex algebraic orbifold groups, their characteristic varieties, and their abelian covers. In particular, we deal with the question of (quasi)-projectivity of orbifold groups. We also prove a structure theorem for the variety of characters of normal-crossing quasi-projective orbifold groups. …

We study the Abel-Jacobi map for bisections of a certain rational elliptic surface. As an application, we construct examples of Zariski $N$-plets for conic arrangements.

Proves unique maps from certain spaces to others.

Character variety of Whitehead link described in detail.

In this paper we investigate Uludag's method for constructing new curves whose fundamental groups are central extensions of the fundamental group of the original curve by finite cyclic groups. In the first part, we give some generalizations to his method in order to get new families of curves with controlled fundamenta…

The study explores deformations of discrete subgroups in non-compact homogeneous spaces.

Hyperbolic groups' infinite orbits spread evenly in spaces.

Study of translation covers of platonic solids reveals monodromy group structures.

Complex projective manifolds without rational curves are quotients of Abelian varieties.

Odd-dimensional SL(n,Q) contains dense surface subgroups.