# Questions tagged [positive-characteristic]

The positive-characteristic tag has no usage guidance.

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### 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^...

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727 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 ...

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442 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}...

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49 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 ...

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147 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 ...

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117 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 ...

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163 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 $...

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296 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 ...

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458 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 ...

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137 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, ...

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281 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^{...

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181 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 $K := k(x)$ ...

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212 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'...

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102 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. ...

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180 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 ...

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### 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,...

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### 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 ...

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291 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,\...

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240 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 ...

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461 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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290 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 $\mathbb{F}_q$, $q=p^m$, $...

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204 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 ...

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269 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 ...

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### 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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147 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 ...

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244 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 ...

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425 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{...

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229 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?
...

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153 views

### Can I combine the category of Drinfeld modules and the category of the base O_S

I am learning about Drinfeld modules,T-modules,...They are said to be analogues of elliptic curves, abelian varieties,...
Let K be a finite extension of k = Frac(A), and $O_K$ the integral closure of ...

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### 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 ...

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### 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 $\ell$. ...

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229 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 ...

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699 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 ...

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### 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, ...

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495 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, ...

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790 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 ...

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270 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?

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538 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 ...

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632 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 ...

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158 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 ...

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### Excellent schemes

I noticed that many results in positive characteristic assumes that the object of the theorem is excellent. I have looked up the definition of excellent and have tried to get a feeling for it, but all ...

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### On a proposition in Hartshorne's paper “Ample vector bundles on curves”

In Prop. 4.1, p. 87 of the article "Ample vector bundles on curves" (Nagoya Math. J. 43 [1971], 73--89), R. Hartshorne states the following:
Let $A$ be an abelian variety [over an alg. closed field $...

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201 views

### Picard scheme of varieties over imperfect fields

Let $k$ be a field and $X$ a proper $k$-scheme. It is a theorem of Murre and Oort that the Picard functor is representable by a $k$-group scheme $\operatorname{Pic}_{X/k}$ which is locally of finite ...

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694 views

### Are there noncommutative extensions of $\alpha_p$ by $\mathbb{G}_m$?

Let $k$ be a field of characteristic $p > 0$ (algebraically closed, if you want; that doesn't make a difference). Consider a finite type $k$-group scheme $E$ that is a (central) extension of $\...

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### Deligne's exterior power

In "Catégories Tannakiennes", Deligne defines the $n$th exterior power of an object $A$ of an abelian tensor category $\mathcal{C}$ as the image of the morphism
$$p : A^{\otimes n} \to A^{\otimes n}, ...

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527 views

### Can a division algebra have degree divisible by its characteristic?

I apologize in advance if this is easy, but I've tried Googling, and had no luck.
I'm currently working on a proof, and I realized in the course of writing that this proof will break if out there ...

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325 views

### Lifting to char 0, references and questions

Suppose that I have a surface $S$, smooth proper over an algebraically closed (perfect?)field $k$ that lifts algebraically to some $S_W$ defined over a field of char 0. I am interested in properties ...

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903 views

### Number of curves over a finite field

Let $K$ be a finite field. Is there a formula for the number of isomorphism classes of genus $g$ smooth curves over $K$?
In other words does there exists a formula for the number of rational points ...

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517 views

### Nagata's conjecture in positive characteristic

For a $\mathbb C P^2$ is known a result: if through the generic points $p_1,p_2,\dots p_n$ with multiplicities $m_1,m_2\dots, m_n$ correspondingly a degree $d$ irreducible reduced curve passes then $d^...

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611 views

### Does every linear group admit a subgroup of dimension 1?

Suppose that $G$ is a linear group of positive dimension, defined over some field $k$. Is that true, that $G$ admits a (closed) one-dimensional subgroup?
I'm pretty much sure this is true in ...