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In this paper, we obtain a characterizations of the recurrence of a continuous vector field $w$ of a closed connected surface $M$ as follows. The following are equivalent: 1) $w$ is pointwise recurrent. 2)$w$ is pointwise almost periodic.…

Let G be a reductive algebraic group in classical types A, B, D and e be an element of its Lie algebra with Z its centraliser in G for the adjoint action. We suppose that e identifies with an nilpotent matrix of order two, which guarantees…

Consider all moduli points corresponding with polarized abelian varieties in characteristic p such that the associated quasi-polarized p-divisible group is geometrically isomorphic with a given one. This defines a subset C of the moduli…

For a Lie groupoid $\mathcal{G}$ with Lie algebroid $A$, we realize the symplectic leaves of the Lie-Poisson structure on $A^*$ as orbits of the affine coadjoint action of the Lie groupoid $\mathcal{J}\mathcal{G}\ltimes T^*M$ on $A^*$,…

Dual representations $V$ and $V^*$ of a complex connected algebraic group $G$ simultaneously have either infinitely or finitely many orbits. Whenever the latter holds, the orbits in $V$ and $V^*$ are in a bijective correspondence called…

A projective symplectic variety $\mathcal{P}$ of dimension 6, with only finite quotient singularities, $\pi(\mathcal{P})=0$ and $h^{(2,0)}(\mathcal{P}_{smooth})=1$, is described as a relative compactified Prym variety of a family of genus 4…

In this paper we prove birational superrigidity of finite covers of degree $d$ of the $M$-dimensional projective space of index 1, where $d\geqslant 5$ and $M\geqslant 10$, with at most quadratic singularities of rank $\geqslant 7$,…

In this article parametric versions of Wilson's plug and Kuperberg's plug are discussed. We show that there is a weak homotopy equivalence induced by the inclusion between the space of non-singular vector fields tangent to a foliation and…

Given the integral lattice $\Lambda^d$ in $d$-dimensional Euclidean space, partitions of the lattice nodes into orbits of finite-index subgroups of $Aut(\Lambda^d)$ have been computed for $d \leq 4$. These partitions can be interpreted as…

We classify fibrations by integral plane projective rational quartic curves whose generic fibre is regular but admits a non-smooth point that is a canonical divisor. These fibrations can only exist in characteristic two. The geometric…

Let $U_\epsilon(\mathfrak g)$ be the simply connected quantized enveloping algebra associated to a finite-dimensional complex simple Lie algebra $\mathfrak g$ at the roots of unity. The De Concini-Kac-Procesi conjecture on the dimension of…

Let F be a holomorphic foliation of P^2 by Riemann surfaces. Assume all the singular points of F are hyperbolic. If F has no algebraic leaf, then there is a unique positive harmonic $(1,1)$ current $T$ of mass one, directed by F. This…

We characterise smooth curves in a smooth cubic threefold whose blow-ups produce a weak-Fano threefold. These are curves $C$ of genus $g$ and degree $d$, such that (i) $2(d-5) \le g$ and $d\le 6$; (ii) $C$ does not admit a 3-secant line in…

We give a complete classification of the finite $2$-groups $G$ for which the automorphism group $\operatorname{Aut}(G)$ acting naturally on $G$ has three orbits. There are two infinite families and one additional group, of order $2^9$. All…

We study the problem of classifying the lines of the projective $3$-space $PG(3,q)$ over a finite field $GF(q)$ into orbits of the group $G=PGL(2,q)$ of linear symmetries of the twisted cubic $C$. A generic line neither intersects $C$ nor…

We exhibit an explicit formula for the cardinality of solutions to a class of quadratic matrix equations over finite fields. We prove that the orbits of these solutions under the natural conjugation action of the general linear groups can…

Working over an algebraically closed field of characteristic p > 3, we calculate the orbit closures in the Witt algebra W under the action of its automorphism group G. We also outline how the same techniques can be used to determine…

Let F be the flag variety of a complex semi-simple group G, let H be an algebraic subgroup of G acting on F with finitely many orbits, and let V be an H-orbit closure in F. Expanding the cohomology class of V in the basis of Schubert…

It is known after Jouanolou that a general holomorphic foliation of degree $\geq2$ in projective space has no algebraic leaf. We give formulas for the degrees of the subvarieties of the parameter space of one-dimensional foliations that…

It is known that symmetric orbits in ${\bf g}^*$ for any simple Lie algebra ${\bf g}$ are equiped with a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the reduced Sklyanin bracket associated to the "canonical"…