Let K be a complete field of unequal characteristics $(0,p)$. The aim of this paper is to describe the the semi-stable models for covers $\bold P^1_K@>>>\bold P^1_K$ of degree p, unramified outside $r\leq p$ points and totally ramified above one of them, under the assumption that the ramified locus has a particular reduction type (which always occurs if $r\leq4$). We are principally concerned with the minimal semi-stable models which separate the ramified fibers.
The central topic is this question: is a given $k$-étale algebra $\prod_lE_l/k$ the specialization of a given $k$-cover $f:X\rightarrow B$ at some point $t_0\in B(k)$? Our main tool is a {\it twisting lemma} that reduces the problem to finding $k$-rational points on a certain $k$-variety. Previous forms of this twisting lemma are generalized and unified. New applications are given: a Grunwald form of Hilbert's irreducibility theorem over number fields, a non-Galois variant of the Tchebotarev theorem for function fields over finite fields, some general specialization properties of covers over P…
The paper has three main applications. The first one is this Hilbert-Grunwald statement. If $f:X\rightarrow \Pp^1$ is a degree $n$ $\Qq$-cover with monodromy group $S_n$ over $\bar\Qq$, and finitely many suitably big primes $p$ are given with partitions $\{d_{p,1},..., d_{p,s_p}\}$ of $n$, there exist infinitely many specializations of $f$ at points $t_0\in \Qq$ that are degree $n$ field extensions with residue degrees $d_{p,1},..., d_{p,s_p}$ at each prescribed prime $p$. The second one provides a description of the se-pa-ra-ble closure of a PAC field $k$ of characteristic $p\not=2$: it is ge…
This paper is a continuation of an earlier paper of the authors on the probem of specializations of modules. The aim here is to define specialisations of finitely generated modules over local rings of the form k(u)[X]_P, where u is a family of indeterminates over a field k. The definition depends on the associated primes of the specialisation of the prime ideal P by the substitution of u to elements of k. As in the non-local case, the introduced specializations preserve basic properties and operations on modules.In particular, annihilators of local cohomology modules commute with specializatio…
Let $E$ be an elliptic surface over the curve $C$, defined over a number field $k$, let $P$ be a section of $E$, and let $\ell$ be a rational prime. For any non-singular fibre $E_t$, we bound the number of points $Q$ on $E_t$ of (algebraic) degree at most $D$ over $k$, such that $\ell^n Q=P_t$, for some $n\geq 1$. The bound obtained depends only on $\ell$, the surface and section in question, $D$, and the degree $[k(t):k]$; that is, it is uniform across all fibres of bounded degree. In special cases, we obtain more specific, in some instances sharp, bounds.
We address some questions concerning indecomposable polynomials and their behaviour under specialization. For instance we give a bound on a prime $p$ for the reduction modulo $p$ of an indecomposable polynomial $P(x)\in \Zz[x]$ to remain indecomposable. We also obtain a Hilbert like result for indecomposability: if $f(t_1,...,t_r,x)$ is an indecomposable polynomial in several variables with coefficients in a field of characteristic $p=0$ or $p>°(f)$, then the one variable specialized polynomial $f(t_1^\ast+α_1^\ast x,...,t_r^\ast+α_r^\ast x,x)$ is indecomposable for all $(t_1^\ast, ..., t_r^\a…
The paper is concerned with the following version of Hilbert's irreducibility theorem: if $π: X \to Y$ is a Galois $G$-covering of varieties over a number field $k$ and $H \subset G$ is a subgroup, then for all sufficiently large and sufficiently divisible $n$ there exist a degree $n$ closed point $y \in |Y|$ and $x \in π^{-1}(y)$ for which $k(x)/k(y)$ is a Galois $H$-extension, and $k(y)/k$ is an $S_n$-extension. The result has interesting corollaries when applied to moduli spaces of various kinds. For instance, for every finite group $G$ there is a constant $N$ such that for all $n>N$ there …
We study the behavior of theta characteristics on an algebraic curve under the specialization map to a tropical curve. We show that each effective theta characteristic on the tropical curve is the specialization of $2^{g-1}$ even theta characteristics and $2^{g-1}$ odd theta characteristics. We then study the relationship between unramified double covers of a tropical curve and its theta characteristics, and use this to define the tropical Prym variety.
Suppose that $K$ is a field of characteristic 0, $K_a$ is its algebraic closure, $p$ is a prime, $q=p^r$ is a power prime. Suppose that $f(x) \in K[x]$ is a polynomial of degree $n > 4$ without multiple roots. Let us consider the superelliptic curve $C: y^q=f(x)$ and its jacobian $J(C)$. We study the endomorphism algebra $End^0(J(C))$ of all $K_a$-endomorphisms of $J(C)$. We prove that $End^0(J(C))$ is "as small as possible" if the Galois group of $f$ over $K$ is either the full symmetric group $S_n$ or the alternating group $A_n$.
Let G be a connected algebraic group over an algebraically closed field of characteristic p (possibly 0), and X a variety on which G acts transitively with connected stabilizers. We show that any étale Galois cover of X of degree prime to p is also homogeneous, and that the maximal prime-to-p quotient of the étale fundamental group of X is commutative. We moreover obtain an explicit bound for the number of topological generators of the said quotient. When G is commutative, we also obtain a description of the prime-to-p torsion in the Brauer group of G.
Let $R$ be a complete discrete valuation ring of equal characteristic $p>0$. Given a $\mathbb{Z}/p$-Galois cover of a formal disc over $R$, one can derive from it a semi-stable model for which the specializations of branch points are distinct and lie in the smooth locus of the special fiber. The description leads to a combinatorial object which resembles a classical Hurwitz tree in mixed characteristic, which we will give the same name. The existence of a Hurwitz tree is necessary for the existence of a $\mathbb{Z}/p$-cover whose branching data fit into that tree. We show that the conditions i…
Let $K$ be a field, let $G$ be a finite group, and let $\bar X\rightarrow \bar Y$ be a $G$-Galois branched cover of varieties over $K^{sep}$. Given a mere cover model $X\rightarrow Y$ of this cover over $K$, in Part I of this paper I observe that there is a unique minimal field $E$ over which $X\rightarrow Y$ becomes Galois, and I prove that $E/K$ is Galois with group a subgroup of $Aut(G)$. In Part II of this paper, by making the additional assumption that $K$ is a field of definition (i.e., that there exists \it some \rm Galois model over $K$), I am able to give an explicit description of th…
Let $p$ be a prime and $q$ be a power of $p$. We compute the Chow ring of the classifying space of some Chevalley groups $G(\mathbb{F}_q)$, when considered as a finite algebraic group over a field of characteristic $p$ containing $\mathbb{F}_q$. Using specialization from characteristic $0$ to characteristic $p$ we also obtain results over the complex numbers.
Inspired by recent work of Farb, Kisin and Wolfson, we develop a method for using actions of finite group schemes over a mixed characteristic dvr R to get lower bounds for the essential dimension of a cover of a variety over K = Frac(R). We then apply this to prove p-incompressibility for congruence covers of a class of unitary Shimura varieties for primes p at which the reduction of the Shimura variety (at any prime of the reflex field over p) does not have any ordinary points. We also make some progress towards a conjecture of Brosnan on the p-incompressibility of the multiplication by p map…
We study the $p$-rank stratification of the moduli space of cyclic degree $\ell$ covers of the projective line in characteristic $p$ for distinct primes $p$ and $\ell$. The main result is about the intersection of the $p$-rank $0$ stratum with the boundary of the moduli space of curves. When $\ell=3$ and $p \equiv 2 \bmod 3$ is an odd prime, we prove that there exists a smooth trielliptic curve in characteristic $p$, with every genus $g$, signature type $(r,s)$ and $p$-rank $f$ satisfying the clear necessary conditions.
We give a comprehensive treatment on how $F$-signatures, splitting primes, splitting ratios, and test modules behave under finite covers. To this end, we expand on the notion of transposability along a section section of the relative canonical module as first introduced by K.~Schwede and K.~Tucker.
We study the deformation of $G$-marked stable curves in the case where $G$ is a cyclic group, and construct a parameterizing space for $G$-marked stable curves of a given numerical type. This is then used in order to study the components of the locus of stable curves admitting the action of a cyclic group of non prime order $d$, extending work of F. Catanese in the case where $d$ is prime.
EPW-sextics are special 4-dimensional sextic hypersurfaces (with 20 moduli) which come equipped with a double cover. We analyze the double cover of EPW-sextics parametrized by a certain prime divisor in the moduli space. We associate to the generic sextic parametrized by that divisor a K3 surface of genus 6 and we show that the double epw sextic is a contraction of the Hilbert square of the K3. This result has two consequences. First it gives a new proof of the following result of ours: smooth double EPW-sextics form a locally complete family of hyperkaehler projective deformations of the Hilb…
We study the homogeneous artinian ideals of the polynomial ring $K[x,y,z]$, generated by the homogenous polynomials of degree $d$ which are invariant under an action of the cyclic group $\mathbb Z/d\mathbb Z$, for any $d\geq 3$. We prove that they are all monomial Togliatti systems, and that they are minimal if the action is defined by a diagonal matrix having on the diagonal $(1, e, e^a)$, where $e$ is a primitive $d$-th root of the unity. We get a complete description when $d$ is prime or a power of a prime. We also establish the relation of these systems with linear Ceva configurations.
