Topological type of discriminants of some special families

We will describe the topological type of the discriminant curve of the morphism $$(\ell , f)$$ ( ℓ , f ) , where $$\ell $$ ℓ is a smooth curve and f is an irreducible curve (branch) of multiplicity less than five or a branch such that the difference between its Milnor number and Tjurina number is less than 3. We prove that for a branch of these families, the topological type of the discriminant curve is determined by the semigroup, the Zariski invariant and at most two other analytical invariants of the branch.

Dimension of the Moduli Space of a Germ of Curve in ℂ2

In this article, we prove a formula that computes the generic dimension of the moduli space of a germ of irreducible curve in the complex plane. It is obtained from the study of the Saito module associated to the curve, which is the module of germs of holomorphic $1$-forms leaving the curve invariant.

The Dynamical Mordell–Lang Conjecture for Skew-Linear Self-Maps. Appendix by Michael Wibmer

Let $k$ be an algebraically closed field of characteristic $0$, let $N\in{\mathbb{N}}$, let $g:{\mathbb{P}}^1{\longrightarrow } {\mathbb{P}}^1$ be a nonconstant morphism, and let $A:{\mathbb{A}}^N{\longrightarrow } {\mathbb{A}}^N$ be a linear transformation defined over $k({\mathbb{P}}^1)$, that is, for a Zariski-open dense subset $U\subset{\mathbb{P}}^1$, we have that for $x\in U(k)$, the specialization $A(x)$ is an $N$-by-$N$ matrix with entries in $k$. We let $f:{\mathbb{P}}^1\times{\mathbb{A}}^N{\dashrightarrow } {\mathbb{P}}^1\times{\mathbb{A}}^N$ be the rational endomorphism given by $(x,y)\mapsto (\,g(x), A(x)y)$. We prove that if $g$ induces an automorphism of ${\mathbb{A}}^1\subset{\mathbb{P}}^1$, then each irreducible curve $C\subset{\mathbb{A}}^1\times{\mathbb{A}}^N$ that intersects some orbit $\mathcal{O}_f(z)$ in infinitely many points must be periodic under the action of $f$. Furthermore, in the case $g:{\mathbb{P}}^1{\longrightarrow } {\mathbb{P}}^1$ is an endomorphism of degree greater than $1$, then we prove that each irreducible subvariety $Y\subset{\mathbb{P}}^1\times{\mathbb{A}}^N$ intersecting an orbit $\mathcal{O}_f(z)$ in a Zariski dense set of points must be periodic. Our results provide the desired conclusion in the Dynamical Mordell–Lang Conjecture in a couple new instances. Moreover, our results have interesting consequences toward a conjecture of Rubel and toward a generalized Skolem–Mahler–Lech problem proposed by Wibmer in the context of difference equations. In the appendix it is shown that the results can also be used to construct Picard–Vessiot extensions in the ring of sequences.

Connected Numbers and the Embedded Topology of Plane Curves

The splitting number of a plane irreducible curve for a Galois cover is effective in distinguishing the embedded topology of plane curves. In this paper, we define the connected number of a plane curve (possibly reducible) for a Galois cover, which is similar to the splitting number. By using the connected number, we distinguish the embedded topology of Artal arrangements of degree b ≥ 4, where an Artal arrangement of degree b is a plane curve consisting of one smooth curve of degree b and three of its total inflectional tangents.

The group of automorphisms of the moduli space of principal bundles with structure group $F_4$ and $E_6$

Let $X$ be a smooth complex projective irreducible curve of genus $g \geq 3$. Let $G$ be the simple complex exceptional Lie group $F_4$ or $E_6$ and let $M(G)$ be the moduli space of principal $G$-bundles. In this work we describe the group of automorphisms of $M(G)$. In particular, we prove that the only automorphisms of $M(F_4)$ are those induced by the automorphisms of the base curve $X$ by pull-back and that the automorphisms of $M(E_6)$ are combinations of the action of the automorphisms of $X$ by pull-back, the action of the only nontrivial outer involution of $E_6$ on $M(E_6)$ by taking the dual and the action of the third torsion of the Picard group of $X$ by tensor product. We also prove a Torelli type theorem for the moduli spaces of principal $F_4$ and $E_6$-bundles, which we use as an auxiliary result in the proof of the main theorems, but which is interesting in itself. We finally draw some conclusions about the way we can see the natural map $M(F_4) \rightarrow M(E_6)$ induced by the inclusion of groups $F_4 \hookrightarrow E_6$.

CERTAIN GENERALIZED MORDELL CURVES OVER THE RATIONAL NUMBERS ARE POINTLESS

A conjecture of Scharaschkin and Skorobogatov states that there is a Brauer–Manin obstruction to the existence of rational points on a smooth geometrically irreducible curve over a number field. In this paper, we verify the Scharaschkin–Skorobogatov conjecture for explicit families of generalized Mordell curves. Our approach uses standard techniques from the Brauer–Manin obstruction and the arithmetic of certain threefolds.

Remarks on Murre's conjecture on Chow groups

For certain product varieties, Murre's conjecture on Chow groups is investigated. More precisely, let k be an algebraically closed field, X be a smooth projective variety over k and C be a smooth projective irreducible curve over k with function field K. Then we prove that if X (resp. XK) satisfies Murre's conjectures (A) and (B) for a set of Chow-Künneth projectors {, 0 ≤ i ≤ 2dim X} of X (resp. for {()K} of XK) and if for any j, , then the product variety X × C also satisfies Murre's conjectures (A) and (B). As consequences, it is proved that if C is a curve and X is an elliptic modular threefold over k (an algebraically closed field of characteristic 0) or an abelian variety of dimension 3, then Murre's conjecture (B) is true for the fourfold X × C.