All Questions
385 questions
37
votes
3
answers
5k
views
Is there a nice proof of the fact that there are (p-1)/24 supersingular elliptic curves in characteristic p?
If $k$ is a characteristic $p$ field containing a subfield with $p^2$ elements (e.g., an algebraic closure of $\mathbb{F}_p$), then the number of isomorphism classes of supersingular elliptic curves ...
- 45.7k
27
votes
4
answers
3k
views
Have people successfully worked with the full ring of differential operators in characteristic p?
This question is inspired by an earlier one about the possibility of using the full ring of differential operators on a flag variety to develop a theory of localization in characteristic $p$. (Here ...
- 57.6k
13
votes
0
answers
749
views
Rings whose Frobenius is flat
Let $R$ be a ring of characteristic $p>0$. The (absolute) Frobenius is the map of rings $F_R:R\rightarrow R$ defined by $x\mapsto x^p$.
I am interested in rings for which $F_R$ is flat (hence ...
anon1432
19
votes
1
answer
977
views
Lang's Jacobian identity: slicker, elementary proof?
In Jeffrey Lang, A Jacobian identity in positive characteristic, J. Commut. Algebra, Volume 7, Number 3 (2015), pp. 393--409, the following result is proven:
Theorem 1. Let $p$ be a prime. Let $\...
- 33.8k
4
votes
2
answers
460
views
Are Chow groups invariant under universal homeomorphisms?
Let $f:X\to Y$ be a universal homeomorphism of schemes, $R$ a coefficient ring. Which assumptions on $f$ and $R$ are suffient to ensure that the pullback map $f^*$ of $R$-linear Chow groups is ...
- 16.9k
12
votes
2
answers
883
views
Residues of $\frac{1}{\prod_{i=1}^n (x-P_i)^{e_i}}$
This is a problem occurring in my research about deformations of $\mathbb{Z}/p^n$-covers over a ring of power series. Given an algebraically closed field $k$ of characteristic $p>0$, suppose $1< ...
- 245
15
votes
2
answers
596
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}...
- 233
15
votes
6
answers
3k
views
Characteristic zero and characteristic $p$ in algebraic geometry
Are there non-trivial (i.e. excluding concepts that can be defined only for $p>0$) statements in algebraic geometry that hold for all fields of characteristic $p$ for all prime $p$ but are known ...
user16974
4
votes
1
answer
269
views
Fourier transform on finite groups in characteristic $p>0$
Is there a Fourier theory for finite groups in characteristic $p>0$? Assume that $p$ divides the order $|G|$ of finite groups (or just work with $p$-groups), i.e., in a modular representation-...
- 41
3
votes
0
answers
197
views
How to write down an explicit equation of given degree yielding a smooth hypersurface in a projective space?
Let F be a field of positive characteristic $p$ and let $d,n$ be two positive integers.
Can we explicitly write down an equation defining a smooth hypersurface $X_d⊂\mathbb P^n_F$ of degree d ?
This ...
- 417
3
votes
1
answer
282
views
Frobenius splitting for an excellent, non $F$-finite, $F$-pure hypersurface
Let $p$ be an odd prime. Let $k$ be a field of characteristic $p$ such that $[k:k^p]=\infty$ (i.e. $k$ is not $F$-finite ) .
Also assume that $-1$ is not a square in $k$ . Consider the homogeneous ...
- 642
48
votes
5
answers
15k
views
Algebraically closed fields of positive characteristic
I'm taking introductory algebraic geometry this term, so a lot of the theorems we see in class start with "Let k be an algebraically closed field." One of the things that's annoyed me is that as far ...
- 12.6k
4
votes
0
answers
130
views
Castelnuovo–Mumford regularity and wedge powers in positive characterisitc
A vector bundle on $\mathbb{P}^n$ is said to be $r$-regular if
$$H^i(\mathbb{P}^n,F(r-i))=0$$ for all $i>0$.
It is always true that if $F$ is $r$-regular and $G$ is $s$-regular (both vector bundles)...
- 2,356
9
votes
2
answers
868
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^...
- 1,072
3
votes
1
answer
317
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 ...
- 16.9k
11
votes
2
answers
1k
views
Representations of $\mathrm{SL}(2)$ in characteristic 2
$\DeclareMathOperator\SL{SL}$In characteristic zero one can use the Clebsch-Gordan rule to decompose tensor products of $\SL(2)$-modules. In characteristic $p$, things are more complicated.
I am ...
- 605
3
votes
1
answer
251
views
Do Neron-Severi groups of smooth projective unirational varieties contain $\ell$-torsion?
Let $X$ be a smooth projective unirational variety over an algebraically closed field of characteristic $p>0$, and $\ell\neq p$ a prime. My question: can the Neron-Severi group of $X$ contain (non-...
- 16.9k
11
votes
2
answers
1k
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 ...
- 2,011
3
votes
0
answers
218
views
Is there a proper smooth variety in characteristic $p$ whose Hodge-to-de Rham spectral sequence does not degenerate at $E_1$?
By Deligne-Illusie, such a variety has no lifting to $W_2(k)$. In their paper they state that they do not know if such an example exists. Has this question been answered since then?
- 4,164
24
votes
5
answers
6k
views
Wild Ramification
The question is, loosely put, what is known about wild ramification?
Is there a semi-well-established theory of wild ramification that can be furthered in various specific situations? Or maybe there ...
9
votes
3
answers
2k
views
Characterisation for separable extension of a field
Can someone verify this for me.. or tell me what reference shows me this... is this true:
Let $k$ be a field. Then a field extension $K$ of $k$ is separable over $k$ iff for any field extension $L \...
- 2,275
7
votes
3
answers
927
views
Lefschetz fixed-point theorem for the Frobenius map
Where can one find a proof of Lefschetz fixed-point theorem for the Frobenius map on elliptic curves over algebraic closures of $F_{p}$ ?
This could immediately follow if their coholomogies (for the ...
- 305
17
votes
2
answers
3k
views
Why is one interested in the mod p reduction of modular curves and Shimura varieties?
Why is one interested in the mod p reduction of modular curves and Shimura varieties?
From an article I learned that this can be used to prove the Eichler-Shimura relation which in turn proves the ...
user19475
11
votes
4
answers
3k
views
What does ramification have to do with separability?
Does ramification have anything to do with inseparability? It feels like an extension of Q in which p ramifies should somehow correspond to an extension of F_p(t). Does totally ramified <--> purely ...
- 15.4k
21
votes
2
answers
5k
views
State of resolution in positive characteristic?
Heisuke Hironaka's coming talk makes me wonder how the state of the work on that theme is. So far, I noticed (but didn't read) these papers:
Kawanoue, Hiraku, Toward resolution of singularities over ...
- 10.8k
5
votes
0
answers
148
views
algebraic de Rham cohomology of toric varieties (reference request)
I haven't been able to find anything workable yet, but I'm looking for a reference on the de Rham cohomology of toric varieties, where as many as possible of the following conditions are handled:
...
- 1,142
19
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 ...
- 20.5k
47
votes
2
answers
9k
views
current status of crystalline cohomology?
The great references given on Ilya's question make me wonder about the current status of the many conjectures and open questions in Illusie's survey from 1994 on crystalline cohomology. Obviously (...
5
votes
1
answer
273
views
Singularities of curves that are moving
Let $k$ be an algebraically closed field, let $d\ge 2$ be an integer and let $f,g\in k[x,y,z]$ be two homogeneous polynomials of degree $d$ without common factor.
We want to know what are the ...
- 7,722
5
votes
1
answer
230
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 $$aX^2 + ...
- 103
19
votes
1
answer
2k
views
The Infinitesimal topos in positive characteristic
This question was inspired by and is somewhat related to this question.
In his article "Crystals and the de Rham cohomology of schemes" in the collection "Dix exposes sur la cohomologie ...
- 4,450
11
votes
1
answer
334
views
An identity in Lie algebras over fields of positive characteristic
Let $L$ be a Lie algebra over a field of characteristic $p>0$ and $D$ a derivation of $L$. For every $x\in L$ denote by $\mathrm{ad} x$ the adjoint map $\mathrm{ad}x: L \rightarrow L, a\mapsto [x,...
- 630
34
votes
2
answers
3k
views
The work of E. Artin and F. K. Schmidt on (what are now called) the Weil conjectures.
I was reading Dieudonne's "On the history of the Weil conjectures" and found two things that surprised me. Dieudonne makes some assertions about the work of Artin and Schmidt which are no doubt ...
- 41.4k
28
votes
3
answers
2k
views
Intuitive pictures in characteristic p
This is a tough one, but does anyone know of any images that recall characteristic p geometry (over algebraically closed fields) in some sense? It is not enough if it is some picture that can be also ...
- 2,215
3
votes
0
answers
232
views
Lifting a Frobenius endomorphism under an étale morphism
Let $X$ be a smooth affine scheme over $\mathbb{Z}/{p^2}$ that is a complete intersection, say $X$ is the spectrum of $\mathbb{Z}/{p^2}[x_1,...x_n]/(f_1, ... f_r)$, where $n-r$ is the dimension of $X$....
- 111
3
votes
1
answer
252
views
Is this unipotent group, over characteristic 2, connected?
Let $E_{ij}(x)\in \mathrm{Mat}_{7\times7}(\overline{\mathbb{F}}_2)$ be the matrix with zeros everywhere, except for the value $x$ at $ij$. Set $$a(x)=1+E_{12}(x)+E_{34}(x)+E_{56}(x),\quad b(y)=1+E_{23}...
- 4,593
9
votes
1
answer
1k
views
Coderivations of $S(V)$ correspond to linear maps $S(V) \to V.$ Only over characteristic $0$?
Definition. Let $k$ be a commutative ring. Let $V$ be a $k$-module. We turn the symmetric algebra $\mathrm{S}\left(V\right)$ of $V$ into a graded Hopf algebra by defining the comultiplication
\begin{...
- 33.8k
9
votes
1
answer
1k
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....
- 528
4
votes
0
answers
169
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 ...
- 393
9
votes
1
answer
833
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 ...
- 3,076
8
votes
1
answer
603
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^{...
- 4,070
9
votes
1
answer
381
views
Lifting of families of curves to characteristic 0
Let $k$ be a finite field, $X_0$ be a smooth affine variety over $k$ and $C\rightarrow X$ a smooth projective family of curves of genus $\geq 2$.
By a result of Elkik we can always lift $X_0$ to a ...
- 728
2
votes
0
answers
100
views
Composing equal characteristic and mixed characteristic deformations
Suppose $k$ is an algebraically closed field of characteristic $p>0$ and $A$ is a $k$-point on a moduli space of certain objects over $k$. Suppose, moreover, that $A$ deforms to a $k[[t]]$-point $B$...
- 245
1
vote
0
answers
51
views
Do we have $\cap_{P\in S}\left(\mathcal{O}_P+(1-\tau)(K)\right)=\left(\cap_{P\in S}\mathcal{O}_P\right)+(1-\tau)(K)$?
Let $\mathbb{F}$ be a finite field and let $q$ be its number of elements. Let $(C,\mathcal{O}_C)$ be a geometrically irreducible smooth projective curve over $\mathbb{F}$ and let $K=\mathbb{F}(C)$ be ...
- 1,479
10
votes
6
answers
2k
views
Proofs in the same vein as Ax-Grothendieck
I would like to see other examples of (ideas of) proofs and results in the same vein as the proof of the Ax-Grothendieck theorem. To explain what I mean by "in the same vein", I will quote from the ...
- 23.4k
24
votes
2
answers
2k
views
Have we ever proved any non-solvable case of reciprocity without the Langlands program ?
The reciprocity of the title is the following not completely well-posed problem:
Fix $P(X)$ a monic irreducible polynomial of degree $n$, with coefficients in $\mathbb Z$. "Describe"
(in some sense) ...
- 26.1k
7
votes
2
answers
1k
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 ...
- 3,758
2
votes
1
answer
451
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 ...
- 313
1
vote
0
answers
246
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 $k$-...
10
votes
3
answers
2k
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 ...
- 63.1k