Questions tagged [characteristic-p]

Fields of characteristic $p$, i.e., fields for which there is a prime $p$ such that $px=0$ for each $x$. Do not use this tag for questions on characteristic polynomials of a matrix.

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Shafarevich conjecture for Abelian varieties over global function fields

Let $S$ be a finite set of places of a global function field $K$. Are there finitely many Abelian varieties over $K$ with good reduction outside $S$? What if we exclude isotrivial families?
2 votes
1 answer
88 views

Does a polynomial $P(X,Y)$ that specializes to a polynomial $P(x_0,Y)$ with distinct roots in $\overline{k}$ have distincts roots in $\overline{k(X)}$

Let $k$ be a field, $P\in k[X][Y]$ be a monic polynomial of degree $n$ in $Y$. I am looking for a simple proof of the following fact. "If there exists $x_0\in k$ such that $P(x_0,Y) \in k[Y]$ has ...
3 votes
0 answers
99 views

Nilpotent orbits in characteristic $0$ vs. positive characteristics

Let $G_\mathbb{C}$ be a connected reductive group over $\mathbb{C}$ with Lie algebra $\mathfrak{g}_{\mathbb{C}}$. For any algebraically closed field $k$, let $G_k$ denote the connected reductive group ...
3 votes
0 answers
160 views

Characters of finite fields - Reference request

Let $\mathbf{F}_q$ ($q=p^f$) be a finite field. We are interested in the characters $\chi: \mathbf{F}_q\rightarrow \mathbf{K}$ ($\chi(0)=0$) where the $ \mathbf{K}$ is an alg.closed field of ...
9 votes
0 answers
345 views

How large must the characteristic of $k$ be, for the cohomology of the Lie algebra $\mathfrak{sl}_n(k)$ to be exterior as in characteristic zero?

$\DeclareMathOperator\SU{SU}$In this question, all Lie algebra cohomology is of the form $H^*(\mathfrak{g}; k)$, with $k$ the trivial one-dimensional representation of $\mathfrak{g}$. All Lie algebra ...
2 votes
0 answers
119 views

Classification of restricted Lie algebras of reductive groups

$\DeclareMathOperator\Lie{Lie}$Let $G/K$ be a reductive group over a field $K$. In characteristic $0$ the Lie algebra is invariant under base change of fields, so to understand $\Lie(G)$ it is enough ...
5 votes
1 answer
229 views

Hochschild cohomology of an Azumaya algebra

Let $k$ be a field. Given a commutative $k$-algebra $Z$ and an associative algebra $A$ that is Azumaya over $Z$, do we have an isomorphism of Hochschild cohomologies: $HH^*(A) \cong HH^*(Z)$? This is ...
1 vote
1 answer
114 views

What are the finite étale coverings of a quasi-hyperelliptic surface?

Let $X$ be a quasi-hyperelliptic surface in characteristic 3 where the canonical bundle $K_X$ is trivial. Question: Is there a finite étale covering $Y \rightarrow X$ such that $Y$ is an abelian ...
11 votes
1 answer
364 views

Chromatic representation theory of the symmetric groups?

We know that in characteristic 0, the group ring of the symmetric group $\Sigma_n$ splits via one idempotent for each partition of $n$. In characteristic $p$, I believe the analogous statement is that ...
7 votes
2 answers
976 views

Hochschild Kostant Rosenberg theorem for varieties in positive characteristic?

Is there is a known version of the HKR theorem as proved in say Swan's paper "Hochschild Cohomology of Quasiprojective Varieties" in positive characteristic? I assume something is known about this as ...
2 votes
1 answer
139 views

Finding the mistake in an argument concerning $F$-finite $F$-split local Cohen--Macaulay ring of dimension $1$

Let $R$ be a commutative Noetherian ring, and $\phi: R \to R$ be a ring homomorphism. For an $R$-module $M$, let $^{\phi}M$ be the $R$-module defined via restriction of scalars via $\phi$, i.e., as ...
5 votes
1 answer
128 views

Derived subalgebra of a restricted Lie algebra

Let $L$ be a restricted Lie algebra over a field of characteristic $p>0$. It is well known that the commutator subalgebra $[L,L]$ is not necessarily restricted (that is, closed under the $p$-map). ...
17 votes
0 answers
334 views

Ado's theorem and the reduction to positive characteristic

The synopsis: proofs of Ado theorem in positive characteristic are simple, and in characteristic $0$ are difficult. Can one infer the characteristic $0$ case from the positive characteristic case? The ...
1 vote
0 answers
81 views

Bad primes of twists of modular curves $X_E^{-1}(p)$

I am interested in a good reference for reading about primes of bad reduction for the modular curve $X_E^{-}(N)$, which is a twist of $X(N)$ parametrizing all elliptic curves $E'$ whose $N$-torsion ...
2 votes
1 answer
147 views

Automorphism of positive characteristic field

Suppose $K$ is a field with $\text{char}(K) \geq 0$. Let $L$ be a cyclic extension of $K$ with degree $2n$. We consider the generator $\sigma$ of the Galois group $\text{Gal}(L/K)$. I am interested in ...
4 votes
1 answer
244 views

Relation between positive roots of $E_8$ and $\mathbb{F}_2^8 \setminus \{0\}$

There exists an explicit bijection (due to Cayley, that has built up a very nice table to describe this) between the positive roots of the lattice $E_7$ and $\mathbb{F}_2^6 \setminus \{0\}$ (where $\...
1 vote
0 answers
130 views

Relative compactification without resolutions of singularities

Let $Y$ be a smooth proper variety over a field $k$, let $X$ be a smooth variety over $k$, $U\hookrightarrow X$ the complement of a strict normal crossing divisor and $\phi\colon U\to Y$ a map. By ...
3 votes
0 answers
93 views

What are the possibilities of the general fibres in an Iitaka fibration?

This question is motivated by complex algebraic geometry. If $X$ is a complex algebraic variety with Kodaira dimension in $[1,\dim X-1]$, then the Iitaka fibration (the rational map induced by the ...
4 votes
0 answers
100 views

Boundary divisor of an equivariant compactification of a non-trivial twist of $\mathbb{G}_a^n$

Let $K$ be a non-perfect field of positive characteristic. Then there exist non-trivial forms of $\mathbb{G}_a^n$ which split over finite purely inseparable extensions of $K$, i.e., algebraic groups $...
3 votes
0 answers
88 views

Dimension of a kernel of a cocycle map

Inspired by a previous question (Dimension of a kernel of a linear map) and some of the answers I was given I thought wheter I can generalize the question to the following: Compute the kernel (or at ...
3 votes
0 answers
108 views

Torsion of Fermat hypersurfaces

An interesting invariant of a rationally chain-connected variety $X/k$ is the exponent of the group, $$ \ker{(\mathrm{CH}_0(X_K) \xrightarrow{\deg} \mathbb{Z})} $$ where $K = k(X)$ is the function ...
17 votes
4 answers
2k views

What are supersingular varieties?

For varieties over a field of characteristic $p$, I saw people talking about supersingular varieties. I wanted to ask "why are supersingular varieties interesting". However, as I don't want to ask an ...
3 votes
0 answers
84 views

Obstruction for points to be contained in smooth hypersurfaces in tterms of inseparability degree of residue field

Let $k$ be an imperfect field of char $p>0$ and $x \in \mathbb{P}^n_k$ be closed point of projective space. In this discussion Qing Liu wrote that Over an imperfect field, a reduced point can not ...
18 votes
2 answers
3k views

Bertini theorems for base-point-free linear systems in positive characteristics

Suppose that $X$ is a smooth algebraic variety over an algebraically closed (uncountable if it helps) field of characteristic $p > 0$. Suppose that $L$ is a line bundle, probably ample or at least ...
3 votes
0 answers
104 views

Points with residue fields having big inseparability degree cannot be contained in smooth hypersurfaces

Let $X$ be a $k$-scheme over imperfect field $k$ and $x \in X $ some (reduced) point with residue field $\kappa(x) = \mathcal{O}_{X,x}/ \mathfrak{m}_x$. How to check that if $\kappa(x)$ has "big ...
1 vote
1 answer
78 views

Generic finite subgroups, associated to small finite fields, of reductive algebraic groups

Theorem 1 of [LS] Liebeck and Seitz - On the subgroup structure of exceptional groups says: Let $X = X(q)$ be a quasisimple group of Lie type in characteristic $p$, and suppose that $X < G$, where ...
2 votes
0 answers
112 views

Resolution of singularities of the resultant locus

We consider projective space of dimension $n$ as the parameter space of degree $n$ polynomials in one variable. Then, I am interested in resolving the singularities of the "resultant locus" $...
4 votes
0 answers
69 views

Conjugacy of cocharacters from conjugacy of labelled diagrams

Everything to follow is over some fixed algebraically closed field $k$. Although all the definitions make sense regardless of characteristic, the meat of the question is about small positive ...
2 votes
0 answers
117 views

How to compute the character of the Steinberg module for the group $\mathrm{SL}_n$ over a field of characteristic $p$?

It is known that the Steinberg representation $V$ of the group $\mathrm{SL}_n$ over a field $k$ of characteristic $p$ (maybe one needs to assume that $k$ is perfect, I am not sure) is the irreducible ...
3 votes
0 answers
119 views

Isogeny of elliptic curve over positive characteristic $p$ which does not come from characteristic $0$

Let $K$ be quadratic imaginary field. Let $E$ be an elliptic curve which has CM over $R_K$ ($R_K$ is ring of integers of $K$). According to SIlverman's ''ADvanced topics in the arithmetic of elliptic ...
2 votes
0 answers
44 views

Characters of simple $\mathfrak{sl}_n$-modules in positive characteristic with subregular nilpotent central character

Consider representations of $\mathfrak{sl}_n$ in positive characteristic with a subregular nilpotent central character $\chi$ (i.e. $\chi$ is a nilpotent matrix whose Jordan normal form has two blocks ...
3 votes
0 answers
59 views

An analog of a BGG resolution in subregular case in positive characteristic

Consider representations of $\mathfrak{sl}_n$ in positive characteristic with a subregular nilpotent central character $\chi$. For every regular weight $\lambda$ of $\mathfrak{sl}_n$, we have the ...
3 votes
0 answers
118 views

Resolving the "wild" singularities of $\mathbb A^n/C_n$

Let the cyclic group on $n$ elements, $C_n$, act on $\mathbb A^n$ by permuting the co-ordinates (over a field $k$). If $n \neq 0 \in k$, we can resolve the singularities of $X = \mathbb A^n/C_n$ by ...
1 vote
1 answer
139 views

When is $R$ a direct summand of Frobenius pushforwards?

Let $(R,\mathfrak m)$ be a reduced Noetherian local ring of prime characteristic $p$. For integer $e>0$, let $F^e_* R$ denote the $R$-module which is $R$ as an abelian group, but the $R$-module ...
2 votes
0 answers
162 views

How do characters of representations in cohomology depend on the (positive-characteristic) field?

The following sentence appears in Jantzen - Representations of algebraic groups, 2nd edition, p. x, where $G$ is a reductive group over an algebraically closed field $k$, $B$ is a Borel subgroup, $T$ ...
4 votes
1 answer
200 views

Compute de Rham-Witt sheaves

I am really new to this, but I am having a hard time understanding all the de Rham-Witt construction. It seems to be really difficult to compute anything with those beasts: like I cannot find any ...
9 votes
0 answers
1k views

Ample vector bundles, $H^1=0$ and global generation in characteristic $p$

This is a follow up from Ample vector bundles on curves question, in characteristic $p>0$. First, some background. It is known that in characteristic $0$, a vector bundle $E$ on say a projective ...
5 votes
1 answer
656 views

A regular, geometrically reduced but non-smooth curve

Can anyone give an example of a projective, regular, geometrically reduced but non-smooth curve ? Of course, the base field should be imperfect. In Exercise 4.3.22 of Qing Liu's book Algebraic ...
4 votes
0 answers
197 views

Inequalities between numerical invariants of nonsingular projective Varieties in positive Characteristic

It is well-known that Miyaoka and Yau-type inequalities do not hold in positive characteristic. In "a note on Bogomolov-Gieseker’s inequality in positive characteristic", however, we can ...
2 votes
0 answers
221 views

Künneth formula for algebraic de Rham cohomology

Let $X$ and $Y$ be finite type schemes over a field $k$, and let $H^i(X/k)$ denote the $i$-th algebraic de Rham cohomology group of $X$ over $k$. I'm interested in the extent to which a Künneth ...
3 votes
1 answer
617 views

Decomposition theorem for polarized abelian varieties in positive characteristic

In characteristic zero we have the following decomposition theorem for polarized abelian varieties: it gives an isomorphism between a PPAV and a product of PPAV's of lower dimension and is valid (as ...
3 votes
1 answer
266 views

Is the theta characteristic attached to an etale double cover of a plane quintic arising from a cubic threefold even in characteristic two?

We work over an algebraically closed field $k$ of characteristic $2$. Let $X$ be a cubic threefold realized as a conic bundle via $f: X \to \mathbb{P}^2$ (after blowing up some line). Let $C \to \...
4 votes
0 answers
263 views

modularity of elliptic curves over function fields in positive characteristic

Let $F$ be a global function field over a finite field, and $E$ be an elliptic curve over $F$. Much like the number field case, it is natural to study the Galois representation on the Tate module of $...
2 votes
0 answers
198 views

Borel-Weil-Bott theorem for wonderful compactification in characteristic p

Are there any known results for a Borel-Weil-Bott theorem for the wonderful compactifications over characteristic $p$ (i.e., theorems that classify the cohomologies of all line bundles on a wonderful ...
10 votes
1 answer
1k views

Are there "reasonable" criteria for existence/non-existence of Levi factors or their conjugacy in prime characteristic?

Classical theorems attributed to Levi, Mal'cev, Harish-Chandra for a finite dimensional Lie algebra over a field of characteristic 0 state that it has a Levi decomposition (semisimple subalgebra plus ...
4 votes
0 answers
257 views

de Rham Witt complex vs. de Rham complex of the Witt ring

I am reading the paper "Revisiting the de Rham-Witt complex" by Bhatt-Lurie-Mathew and I am a bit confused about the difference between $W\Omega_R^*$ and $\hat{\Omega}^*_{W(R)}$. Let $\...
3 votes
0 answers
186 views

Explicit description of wonderful compactification for PGL_3

Let $k$ be an algebraically closed field of positive characteristics. Let $X$ be the wonderful compactification of $PGL_3$ (see for example section 6 of "Frobenius Splitting Methods in Geometry ...
2 votes
0 answers
132 views

Automorphism groups of "reductive" Lie algebras in positive characteristic

I put "reductive" in quotes because, of course, in positive characteristic one should speak of Lie algebras of reductive groups, not of reductive Lie algebras. Let $G$ be a reductive group ...
19 votes
5 answers
4k views

Equivalent statements of the Riemann hypothesis in the Weil conjectures

In the cohomological incarnation, the Riemann hypothesis part of the Weil conjectures for a smooth proper scheme of finite type over a finite field with $q$ elements says that: the eigenvalues of ...
2 votes
1 answer
330 views

Lie algebroid in algebraic geometry

When I did net-surfing at home, I met some geometric backgrounds of Lie algebras and encountered the concept of Lie algebroids. In differential geometry, a Lie algebroid seems to be defined as ...

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