Questions tagged [positive-characteristic]

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votes
2answers
104 views

Supersingular elliptic curves and their automorphisms

If $E$ is a supersingular elliptic curve over a finite field of characteristic $p$, what is known about its automorphism group (i.e the stabilizer of a point in algebraic curves terminology). Do all ...
3
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0answers
73 views

Extensions of fraction field and residue field

Let $A \subset B$ be integrally closed local domains, $K(A), K(B)$ be fields of fractions, and $k(A),k(B)$ be residue fields. How to prove $[K(B):K(A)] \ge [k(B):k(A)]$? This question should be easy ...
3
votes
1answer
218 views

What is known about lower etale cohomology of unirational varieties?

Which information is currently known about $H^1_{et}(X,\mathbb{Z}_l)$ and $H^2_{et}(X,\mathbb{Z}_l)$, where $X$ is a smooth unirational variety over an algebraically closed field of finite ...
1
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0answers
108 views

Frobenius twist of a field

Let $k$ be a field of characteristic $p>0$ (not necessarily perfect). Consider the Frobenius endomorphism $F : k \to k$, $x \mapsto x^p$. I am curious about what happens when we take $k$ as a $k$-...
2
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0answers
128 views

What kind of equivalences exist between categories of characteristic $0$ and characteristic $p$?

The tilting equivalence for perfectoid algebras gives an equivalence of categories $$K\text{-perf} \cong K^\flat\text{-perf}$$ where the left-hand-side are algebras in characteristic zero and the ...
2
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0answers
94 views

vanishing of higher algebraic de Rham cohomology and sheaves of differentials for singular curves

I'm looking for some results or references about de Rham cohomology of curves in less-than-optimal cases. The two vanishing results I care about are: $H^{k}_{\text{dR}}(X) = 0$ for $k>2$, since $H^...
5
votes
1answer
151 views

Do regular (but non-smooth) conics over a discretely valued field of characteristic $2$ admit a regular model over the valuation ring?

Let $K$ be a non-perfect field of characteristic $2$. Let $T \subseteq K$ be a discrete valuation ring. Assume there exist $a,b \in K^{\times}$ such that the projective conic $C$ defined by $$aX^2 + ...
4
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0answers
103 views

Non-cyclic Galois groups over the field of formal Laurent series in positive characteristic

This should be an easy question, but I am unfortunately not able to give an answer, so I am sorry if this is not the appropriate level for the site. Let $C$ be an algebraically closed field of ...
4
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0answers
154 views

Fibered surfaces degenerating to Frobenius

Let $R$ be a DVR with algebraically closed, positive characteristic residue field $k$. Let $X\rightarrow Spec(R)$ and $C\rightarrow Spec(R)$ be smooth projective morphisms of relative dimension 2 and ...
10
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1answer
549 views

Gelfand's trick (Gelfand's lemma) in positive characteristic?

I came across this preprint that claims in Lemma 1.1 that Gelfand's trick (also known as Gelfand's lemma) only works in characteristic zero: Let $H < G$ be finite groups. Suppose we have an anti-...
3
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1answer
196 views

Is this unipotent group, over characteristic 2, connected?

Let $E_{ij}(x)\in \mathrm{Mat}_{7\times7}(\overline{\mathbb{F}}_2)$ be the matrix with zeros everywhere, except for the value $x$ at $ij$. Set $$a(x)=1+E_{12}(x)+E_{34}(x)+E_{56}(x),\quad b(y)=1+E_{23}...
9
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2answers
438 views

Nakano vanishing in positive characteristic

Let $X$ be a smooth projective variety defined over a field $k$. In characteristic zero, the following is a special case of the (Kodaira-Akizuki-)Nakano vanishing theorem: $(\ast) \quad$ $\mathrm H^...
15
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2answers
981 views

Biggest Field Of Characteristic $p$

The Surreal nummbers, $\boldsymbol{No}$, are according to Wikipidia the biggest ordered field, and the Surrcomplex numbers are the biggest field of characteristic 0. Biggest in the sense that every ...
15
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2answers
495 views

When is the etale cohomology of $\mathrm{Sym}^n(X)$ isomorphic to the $\Sigma_n$-invariants in the étale cohomology of $X^n$?

Suppose $X$ is a smooth projective variety defined over an arbitrary algebraically closed field $k$, and consider the action of $\Sigma_n$ on the $n$-fold product $X^n$. Is it true that $H_{\acute{e}t}...
4
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0answers
71 views

Symmetric power contained in tensor power?

Let $V$ be an $R$-module. Traditionally the symmetric algebra $S(V)$ is defined as a quotient of the tensor algebra $T(V)$, by the ideal generated by all $a\otimes b-b\otimes a$. Can $S^n(V)$ also be ...
2
votes
1answer
243 views

Tie-Breaking Trick for Log Canonical Pairs and F-pure pairs in Positive Characteristic

Let $X$ be a projective 3-fold in characteristic $p>0$. Let $(X, D)$ be a klt pair, and $D'$ a $\mathbb{R}$-Cartier divisor such that $D'=A'+B'$, where $A\geq 0$ is an ample $\mathbb{Q}$-divisor ...
4
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0answers
137 views

Semisimplicity of the p-adic étale Tate module over $F_p(t)$

Let $k$ be a finitely generated field of positive characteristic p. Let $A$ be an abelian variety over $k$ and write $T_p(A)$ for the $p$-adic étale Tate module of $A$. Is it known if the natural ...
2
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1answer
169 views

Subschemes in group action

Let $k$ be a field, of any characteristic. Let $G$ be a smooth group scheme over $k$ and let $X$ be a smooth scheme of finite type over $k$. Let $Y\subseteq X$ be a smooth subscheme of $X$, and let $...
9
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1answer
463 views

Endomorphism ring of simple ordinary abelian variety

Is there an example of an ordinary and simple abelian variety $A$ over an algebraically closed field $K$ (of characteristic $p>0$) such that ${\rm End}(A)$ is not commutative? Note that the answer ...
11
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2answers
629 views

Cotangent complex of perfect algebra over a perfect field

Let $A$ be a perfect $\kappa$-algebra over a perfect field $\kappa$ of positive characteristic $p$. Then the algebraic (= classical) cotangent complex $L_{A/\kappa}^{\operatorname{alg}}$ is known to ...
5
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0answers
153 views

Map associated to linear system onto curve is morphism

In Mumford's first paper on Surfaces in char $p$ [1], part 2 Step (II), he wants to show that, given an indecomposable curve of canonical type $D$ on a smooth projective surface $F$ with $p_g(F)=0, ...
7
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1answer
362 views

absolute Galois group of the function field of a curve over $\mathbb{F}_p^{alg}$

In their 2008 paper "Torelli theorem for curves over finite fields" Bogomolov, Korotiaev and Tschinkel mention in the beginning of Section 9 that absolute Galois groups of curves over $\mathbb{F}_p^{...
1
vote
1answer
207 views

Norms of elements in Artin-Schreier extensions

The following is claimed in the proof of Theorem 7.5 of Auslander, Goldman, "The Brauer group of a commutative ring": Let $k$ be a nonperfect field of positive characteristic $p$, let $K := k(x)$ ...
2
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0answers
230 views

Surjectivity of map of Picard schemes implies abelian

Note: This question was asked on MSE first, but got zero reactions. So I deleted it there, and am now posting it here. I am looking for a reference or explanation of the fact that is used in Mumford'...
2
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0answers
109 views

Global invariant cycles in positive characteristic

Let $k$ be an algebraically closed field of positive characteristic $p$ and fix a prime $\ell\neq p$. Let $X$ be a smooth connected $k$-variety and $f:Y\rightarrow X$ a smooth projective morphism. ...
3
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0answers
195 views

Isotrivial factors of Jacobian

Let $k$ be an algebraically closed field of positive characteristic that it is not the algebraic closure of a finite field. Fix a smooth proper $k$-curve $C$ and write $J_C$ for its Jacobian abelian ...
10
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1answer
267 views

An identity in Lie algebras over fields of positive characteristic

Let $L$ be a Lie algebra over a field of characteristic $p>0$ and $D$ a derivation of $L$. For every $x\in L$ denote by $\mathrm{ad} x$ the adjoint map $\mathrm{ad}x: L \rightarrow L, a\mapsto [x,...
3
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0answers
100 views

Cohomologies of $[V/GL_n]$ in characteristic $p$ for a representation $V$ of $GL_n$

Let $V$ be a representation of $G=GL_n$ (or more generally any reductive group $G$) over an algebraically closed field $\mathbb k$ of characteristic $p$. Let $[V/G]$ be the corresponding quotient ...
5
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0answers
362 views

Eisenbud-Goto conjecture in Positive Characteristic

Eisenbud-Goto conjecture predicted that the Castelnuovo-Mumford regularity ${\rm reg}(X)$ of a non-degenerate projective variety $X\subset \mathbb{P}^N$ is bounded by the $\deg(X)-{\rm codim}(X,\...
7
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0answers
299 views

Sheaf whose singular support is not Lagrangian

For constructible sheaves $\mathcal F$ on real analytic manifolds $X$, there is a notion of the singular support $SS(\mathcal F)$ which is a radially invariant singular Lagrangian subset of the ...
7
votes
1answer
604 views

deformation theory in positive characteristic

The idea "Formal deformation theory in characteristic zero is controlled by a differential graded Lie algebra (dgla)" goes back to Goldman-Millson, Deligne, Drinfeld among others; see Lurie's ICM talk....
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0answers
324 views

rigid analytic geometry positive characteristic

I am a beginning graduate student. I have the following basic question I am very confused about: Suppose $C$ is a smooth geometrically irreducible curve over a finite field $\mathbb{F}_q$, $q=p^m$, $...
3
votes
1answer
240 views

Derivations of central extensions of simple Lie algebras

Let $L$ be a finite-dimensional Lie algebra over a field of characteristic zero. It is not difficult to see (and also follows from Theorem 4.4 of [G. Hochschild: Semi-simple algebras and generalized ...
6
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0answers
300 views

Are all stabilizer groups of the co-adjoint action smooth?

Let $k$ be a (non-archimedean) local field of positive characteristic $p$ and $\mathfrak{n}$ be any finite-dimensional nilpotent Lie algebra over $k$ with nilpotence length $l<p$. It is well-known ...
1
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1answer
189 views

locally closed orbits in metric Hausdorff topology

I learned the following fact from Bruhat and Tits's paper "Homomorphismes “abstraits” de groupes algebriques simples" Section 3.18 that Let $k$ be a local field. Suppose that a $k$-group $H$ acts $k$...
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0answers
165 views

Unitary dual of the Heisenberg group over non-archimedean local fields of characteristic two

What is the unitary dual of the Heisenberg group over non-archimedean local fields k of characteristic two? This is well-known for the real Heisenberg group and in fact, when local fields have ...
7
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0answers
276 views

Arbitrarily non-degenerate Hodge to de Rham spectral sequence

It is true that for any $n$ there exists a compact complex manifold which Frolicher spectral sequence does not degenerate at the $n$-th page(https://arxiv.org/pdf/0709.0481.pdf). Does the analogous ...
4
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1answer
460 views

finitness of syntomic/fppf cohomology with coefficients in a finite flat group scheme

Let $X/k$ be a smooth projective variety over a finite field of characteristic $p$ and $\mathscr{A}/X$ be an Abelian scheme. Is then $H^1_\mathrm{SYN}(X,\mathscr{A}[p]) = H^1_\mathrm{fppf}(X,\mathscr{...
4
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1answer
252 views

Finiteness of cohomology with finite coefficients

Let $G$ be a finite abelian group and let $S$ be a variety over $\mathbb{F}_p$. It is natural (I think) to expect that the cohomology group $H^i(S,G)$ is finite. But with respect to which cohomology? ...
5
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0answers
162 views

Symplectic leaves in positive characteristic

I am currently considering a family of filtered algebras over an algebraically closed field of positive characteristic with the property that the associated graded algebra is a finitely generated ...
17
votes
1answer
1k views

Independence of $\ell$ of Betti numbers

When $X$ is a smooth proper variety over $\mathbb F_q$, we know by Deligne's theory of weights that the dimension of $H^i_{\operatorname{\acute et}}(\bar X, \mathbb Q_\ell)$ does not depend on $\ell$. ...
2
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0answers
265 views

Does the numerical equivalence relation coincide with the homological one for 1-cycles (in positive characteristic)?

Is the Grothendieck's Standard Conjecture D (stating that the numerical equivalence relation for algebraic cycles with rational coefficients coincides with the homological one) known to be true for ...
5
votes
1answer
820 views

Generic Smoothness Type of Results in Positive Characteristic

Let $f:X\to Y$ be a surjective morphism between two projective varieties over a field of characteristic $p>0$. Also assume that $f_*\mathcal{O}_X=\mathcal{O}_Y$, and $X$ is smooth. We know that ...
6
votes
2answers
742 views

Lefschetz on étale fundamental group for quasi-projective varieties

If $X$ is a smooth projective variety of dimension at least $3$ over $\mathbb{C}$, Lefschetz's Hyperplane theorem says that for every hyperplane section $H$ $$\pi^1(H)\to\pi^1(X)$$ is an isomorphism, ...
7
votes
2answers
561 views

$p$-torsion of an abelian variety of $p$-rank $0$

Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $A$ be an abelian variety over $k$ such that $A[p](k) = 0$, i.e., such that $A$ has $p$-rank $0$. If I am not mistaken, ...
10
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3answers
973 views

Pullback along Frobenius morphism

Let $X$ be a scheme over a finite field $\mathbb{F}_q$ and let $F : X \to X$ be the absolute Frobenius morphism. If $\mathcal{L}$ is an invertible $\mathcal{O}_X$-module, then there is a natural ...
4
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1answer
287 views

Examples of perfect pseudo algebraically closed fields in positive characteristic

Is there any known example of a perfect pseudo algebraically closed field of positive characteristic containing $\overline{\mathbb{F}_p}$ but is not algebraically closed?
8
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1answer
581 views

Automorphisms of curves in positive characteristic

It is well known that over an algebraically closed field of characteristic zero a general curve (for an open subset of $M_g$) of genus $g\geq 3$ is automorphism-free. Is this result still true over ...
10
votes
1answer
709 views

Non-algebraic K3 surfaces in characteristic $p$

I have a very naive question. Recall that over the field of complex numbers, there exist non-algebraic K3 surfaces. Namely, smooth non-projective simply connected compact complex surfaces with ...
3
votes
1answer
170 views

Interpreting Frobenius pullback as an invariant differential in the case of an elliptic curve

Let $S$ be an affine scheme of characteristic $p > 0$, let $E \rightarrow S$ be an elliptic curve over $S$, and let $F$ denote the absolute Frobenius. Since $E$ is its own $\mathrm{Pic}^0$ there is ...