For a prime $p$, a cyclic-by-$p$ group $G$ and a $G$-extension $L|K$ of complete discrete valuation fields of characteristic $p$ with algebraically closed residue field, the local lifting problem asks whether the extension $L|K$ lifts to characteristic zero. In this paper, we characterize $D_4$-extensions of fields of characteristic two, determine the ramification breaks of (suitable) $D_4$-extensions of complete discrete valuation fields of characteristic two, and solve the local lifting problem in the affirmative for every $D_4$-extension of complete discrete valuation fields of characteristic two with algebraically closed residue field; that is, we show that $D_4$ is a local Oort group for the prime 2.
We prove existence and nonexistence results for certain differential forms in positive characteristic, called {\em good deformation data}. Some of these results are obtained by reduction modulo $p$ of Belyi maps. As an application, we solve the local lifting problem for groups with Sylow $p$-subgroup of order $p$.
The lifting problem that we consider asks: given a smooth curve in characteristic p and a group of automorphisms, can we lift the curve, along with the automorphisms, to characteristic zero? One can reduce this to a local question (the so-called local lifting problem) involving continuous group actions on formal power series rings. In this expository article, we overview much of the progress that has been made toward determining when the local lifting problem has a solution, and we give a taste of the work currently being undertaken. Of particular interest is the case when the group of automor…
We solve the local lifting problem for the alternating group A_4, thus showing that it is a local Oort group. Specifically, if k is an algebraically closed field of characteristic 2, we prove that every A_4-extension of k[[s]] lifts to characteristic zero. As a consequence, every A_4-branched cover of smooth projective curves in characteristic 2 lifts to characteristic zero.
Let $G=D_p$ be the dihedral group of order $2p$, where $p$ is an odd prime. Let $k$ an algebraically closed field of characteristic $p$. We show that any action of $G$ on the ring $k[[y]]$ can be lifted to an action on $R[[y]]$, where $R$ is some complete discrete valuation ring with residue field $k$ and fraction field of characteristic 0.
The local Oort conjecture states that, if G is cyclic and k is an algebraically closed field of characteristic p, then all G-extensions of k[[t]] should lift to characteristic zero. We prove a critical case of this conjecture. In particular, we show that the conjecture is always true when v_p(|G|) \leq 3, and is true for arbitrarily highly p-divisible cyclic groups G when a certain condition on the higher ramification filtration is satisfied.
We study finite groups of automorphisms of the $p$-adic open disk. In particular, we generalize results of Green, Matignon and Henrio from cyclic groups of order $p$ to arbitrary finite groups. As an application, we produce a counterexample to a question of Chinburg, Guralnick and Harbater, concerning the local lifting problem for generalized quaternion groups.
The lifting problem for curves with automorphisms asks whether we can lift a smooth projective characteristic p curve with a group G of automorphisms to characteristic zero. This was solved by Grothendieck when G acts with prime-to-p stabilizers, and there has been much progress over the last few decades in the wild case. We survey the techniques and obstructions for this lifting problem, aiming at a reader whose background is limited to scheme theory at the level of Hartshorne's book. Throughout, we include numerous examples and clarifying remarks. We also provide a list of open questions.
Let $k$ be an algebraically closed field of characteristic $p > 0$. We study obstructions to lifting to characteristic 0 the faithful continuous action $ϕ$ of a finite group $G$ on $k[[t]]$. To each such $ϕ$ a theorem of Katz and Gabber associates an action of $G$ on a smooth projective curve $Y$ over $k$. We say that the KGB obstruction of $ϕ$ vanishes if $G$ acts on a smooth projective curve $X$ in characteristic 0 in such a way that $X/H$ and $Y/H$ have the same genus for all subgroups $H \subset G$. We determine for which $G$ the KGB obstruction of every $ϕ$ vanishes. We also consider anal…
Let $k$ be an algebraically closed field of characteristic $p > 0$. We consider the problem of lifting $p$-cyclic covers of $\Proj_k$ as $p$-cyclic covers of the projective line over some DVR under the condition that the wild monodromy is maximal. We answer positively the question for covers birational to $w^ p-w=tR(t)$ for some additive polynomial $R(t)$.
In the spirit of Fakhruddin (arXiv:math/0212208v1) and Szpiro-Bhatnagar (arXiv:1010.2715v1), we show that for an equicharacteristic complete local ring A, with a given embedding of Spec(A) in the prime spectrum Spec(R) of some complete regular local ring R, any finite self map of A can be lifted to a finite self map of R, keeping the given embedding.
We use vanishing results for sheaf cohomology on Siegel modular varieties to study two lifting problems: (a) When can Siegel modular forms (mod p) be lifted to characteristic zero? This uses and extends previous results for cusp forms by Stroh and Lan-Suh. (b) When is the restriction of Siegel modular forms to the boundary of the moduli space a surjective map? We investigate this question in arbitrary characteristic, generalising analytic results of Weissauer and Arakawa.
Let X be a smooth projective hyperelliptic curve over an algeraically closed field k of prime characteristic p. The aim of this note is to find necessary and sufficient conditions on the automorphism group of the curve X to be lifted to characteristic zero. The results will be generalised for a certain family of curves that we call cyclic curves.
Given a tropical divisor $D$ in the intersection of two tropical plane curves, we study when it can be realized as the tropicalization of the intersection of two algebraic curves, and give a sufficient condition. We show that under a certain condition involving a graph determined by these tropical curves, we can algorithmically find algebraic curves such that the tropicalization of their intersection is $D$.
This article presents a new approach to the unobstructedness result for deformations of Calabi-Yau varieties by introducing the tangent lifting property for a functor on artinian local algebras. The verification that the deformation functor of a Calabi-Yau variety in characteristic 0 fulfills the tangent lifting property uses, just as the verification of the $T^1$-lifting property, the degeneration of the Hodge to de Rham spectral sequence. Our main use of our methods is however to the mixed characteristic case. In that case to be able to verify the conditions needed one needs an extension of …
We give a necessary and sufficient condition for a modular representation of a group $G=C_{p^h} \rtimes C_m$ in a field of characteristic zero to be lifted to a representation over local principal ideal domain of characteristic zero containing the $p^h$ roots of unity.
We construct the moduli space of Enriques surfaces in positive characteristic and eventually over the integers, and determine its local and global structure. As an application, we show lifting of Enriques surfaces to characteristic zero. The key observation is that the canonical double cover of an Enriques surface is birational to the complete intersection of three quadrics in IP^5, even in characteristic 2.
Given a complete real-valued field $k$ of residue characteristic zero, we study properties of a differential form $ω$ on a smooth proper $k$-analytic curve $X$. In particular, we associate to $(X,ω)$ a natural tropical reduction datum combining tropical data of $(X,ω)$ and algebra-geometric reduction data over the residue field $\widetilde{k}$. We show that this datum satisfies natural compatibility condition, and prove a lifting theorem asserting that any compatible tropical reduction datum lifts to an actual pair $(X,ω)$. In particular, we obtain a short proof of the main result of a work [B…
Let $X$ be an integral projective variety of codimension two, degree $d$ and dimension $r$ and $Y$ be its general hyperplane section. The problem of lifting generators of minimal degree $σ$ from the homogeneous ideal of $Y$ to the homogeneous ideal of $X$ is studied. A conjecture is given in terms of $d$, $r$ and $σ$; it is proved in the cases $r=1,2,3$. A description is given of linear systems on smooth plane curves whose dimension is almost maximal.
A classical approach to investigate a closed projective scheme $W$ consists of considering a general hyperplane section of $W$, which inherits many properties of $W$. The inverse problem that consists in finding a scheme $W$ starting from a possible hyperplane section $Y$ is called a {\em lifting problem}, and every such scheme $W$ is called a {\em lifting} of $Y$. Investigations in this topic can produce methods to obtain schemes with specific properties. For example, any smooth point for $Y$ is smooth also for $W$. We characterize all the liftings of $Y$ with a given Hilbert polynomial by a …
Let k be an algebraically closed field of positive characteristic p. We consider which finite groups G have the property that every faithful action of G on a connected smooth projective curve over k lifts to characteristic zero. Oort conjectured that cyclic groups have this property. We show that if a cyclic-by-p group G has this property, then G must be either cyclic or dihedral, with the exception of A_4 in characteristic 2. This proves one direction of a strong form of the Oort Conjecture.
Let f: X -> Z be a local, projective, divisorial contraction between normal varieties of dimension n with Q-factorial singularities. Let $Y \subset X$ be a f-ample Cartier divisor and assume that f|Y: Y -> W has a structure of a weighted blow-up. We prove that f: X -> Z, as well, has a structure of weighted blow-up. As an application we consider a local projective contraction f: X -> Z from a variety X with terminal Q-factorial singularities, which contracts a prime divisor E to an isolated Q-factorial singularity $P\in Z$, such that $-(K_X + (n-3)L)$ is f-ample, for a f-ample Cartier divisor …
Let $k$ be a field with a real valuation $ν$ and $R$ a $k$-algebra. We show that there exist a $k$-algebra $K$ and a real valuation $μ$ on $K$ extending $ν$ such that any real ring valuation of $R$ is induced by $μ$ via some homomorphism from $R$ to $K$; $K$ can be chosen to be a field. Then we study the case when $ν$ is trivial and $R$ a complete local Noetherian ring with the residue field $k$. Let $K$ be the ring $\bar{k}[[t^\R]]$ of Hahn series with its natural valuation $μ$; $\bar{k}$ is an algebraic closure of $k$. Despite $K$ is not universal in the strong sense defined above, it has th…
We study divided power structures on finitely generated $k$-algebras, where $k$ is a field of positive characteristic $p$. As an application we show examples of $0$-dimensional Gorenstein $k$-schemes that do not lift to a fixed noetherian local ring of non-equal characteristic. We also show that Frobenius neighbourhoods of a singular point of a general hypersurface of large dimension have no liftings to mildly ramified rings of non-equal characteristic.
Given $\mathbb P^4_k$, with $k$ algebraically closed field of characteristic $p>0$, and $X\subset \mathbb P^4_k$ integral surface of degree $d$, let $Y=X\cap H$ be the general hyperplane section of $X$. We suppose that $h^0\mathscr I_Y(s)\ne 0$ and $h^0\mathscr I_X(s)=0$ for some $s>0$. This determines a nonzero element $α\in H^1\mathscr I_X(s)$ such that $α\cdot H=0$ in $H^1\mathscr I_X(s)$. We find different upper bounds of $d$ in terms of $s$, $p$ and the order of $α$ and we show that these bounds are sharp. In particular, we see that $d\le s^2$ for $p<s$ and $d\le s^2-s+2$ for $p\ge s$.
