Related papers: Alternating groups as monodromy groups in positive…

In many situations, the monodromy group of enumerative problems will be the full symmetric group. In this paper, we study a similar phenomenon on the rational curves in $|\mathcal{O}(1)|$ on a generic K3 surface of fixed genus over…

Fix a degree $d$ projective curve $X \subset \mathbb{P}^r$ over an algebraically closed field $K$. Let $U \subset (\mathbb{P}^r)^*$ be a dense open subvariety such that every hyperplane $H \in U$ intersects $X$ in $d$ smooth points. Varying…

It is shown that the commutator subgroup of the fundamental group of a smooth affine curve over an uncountable algebraically closed field $k$ of positive characteristic is a profinite free group of rank equal to the cardinality of $k$.

For a curve $X$ of genus $>1$ defined over a finite field, we present a criterion which allows us to state the non existence of automorphisms of order a power of a rational prime. We show how this criterion can be used to determine the…

In positive characteristic, algebraic curves can have many more automorphisms than expected from the classical Hurwitz's bound. There even exist algebraic curves of arbitrary high genus g with more than 16g^4 automorphisms. It has been…

Let k be a field not of characteristic two and L be a set of almost all rational primes invertible in k. Suppose we have a variety X/k and strictly compatible system {M_ell -> X : ell in L} of constructible F_ell-sheaves. If the system is…

In this version small mistakes are corrected and the exposition is changed as suggested by the referee (to appear in Canadian Journal of Mathematics). The first main result of the paper is a criterion for a partially commutative group $\GG$…

A singular curve over a non-perfect field K may not have a smooth model over K. Those are said to "change genus". If K is a global field of positive characteristic and C/K a curve that change genus, then C(K) is known to be finite. The…

Given an integer $\gamma\geq 2$ and an odd prime power $q$ we show that for every large genus $g$ there exists a non-singular curve $C$ defined over $\mathbb{F}_q$ of genus $g$ and gonality $\gamma$ and with exactly $\gamma(q+1)$…

Let X be a smooth variety over an algebraically closed field k of positive characteristic. We define and study a general notion of regular singularities for stratified bundles (i.e. O_X-coherent D_X-modules) on X without relying on…

We consider possibly singular rational projective k*-surfaces and provide an explicit description of the unit component of the automorphism group in terms of isotropy group orders and intersection numbers of suitable invariant curves. As an…

In earlier work, Katz exhibited some very simple one parameter families of exponential sums which gave rigid local systems on the affine line in characteristic p whose geometric (and usually, arithmetic) monodromy groups were SL(2,q), and…

For a quasi-split tamely connected reductive group G over a p-adic field, we prove that its (monodromic) affine Hecke category is canonically equivalent to its equal characteristic counterpart as monoidal categories.

Let $f:\mathbb{K}^n\rightarrow\mathbb{K}^m$ be a generically finite polynomial map of degree $d$ between affine spaces. In arXiv:1411.5011 we proved that if $\mathbb{K}$ is the field of complex or real numbers, then the set $S_f$ of points…

We show the existence of and explicitly construct generic polynomials for various groups, over fields of positive characteristic. The methods we develop apply to a broad class of connected linear algebraic groups defined over finite fields…

Let k be an algebraically closed field. A polynomial F in k[X,Y] is said to be "generally rational" if, for almost all c in k, the curve " F= c '' is rational. It is well known that, if char(k)=0, F is generally rational iff there exists G…

We show that if $K$ is an arbitrary field and $G$ is a finite group then there exists a curve over $K$ with automorphism group $G$. We also give a positive solution to the weak inverse Galois problem for function fields over an arbitrary…

Let $f\colon M\to N$ be a proper map between two aspherical compact orientable 3-manifolds with empty or toroidal boundary. We assume that $N$ is not a closed graph-manifold. Suppose that $f$ induces an epimorphism on fundamental groups. We…

It is proved that there exist plane rational curves of degree twelve (resp. twenty-four) with two different outer Galois points such that the Galois group at one of two Galois points is an alternating group $A_4$ (resp. a symmetric group…

We study the group of automorphisms of the affine plane preserving some given curve, over any field. The group is proven to be algebraic, except in the case where the curve is a bunch of parallel lines. Moreover, a classification of the…