32
votes
State of resolution in positive characteristic?
Also, I'd like to point out this paper of Hironaka...
http://www.math.harvard.edu/~hironaka/pRes.pdf
I haven't read the paper, and also I haven't heard anybody talking about it in the last weeks, ...
29
votes
Accepted
A linear algebra problem in positive characteristic
This is false. Let $c$ be a quadratic nonresidue modulo $p$. Our matrix will be $(p^2-1) \times (p^2-1)$, with rows and colums indexed by pairs $(x,y) \in \mathbb{F}_p^2 \setminus \{ (0,0) \}$.
Our ...
22
votes
Accepted
Biggest Field Of Characteristic $p$
Conway's nimbers form an interesting answer for $p=2$. That every Field of characteristic $2$ embeds into it follows from the fact they form an algebraically closed Field and that they contain ...
19
votes
Biggest Field Of Characteristic $p$
An algebraically closed field is determined up to isomorphism by its characteristic and its transcendence degree over its prime field. So every algebraically closed field of characteristic $p$ is ...
18
votes
Accepted
A short proof for simple connectedness of the projective line
You can deduce this from the classification of vector bundles on $\mathbf{P}^1$. Say $f:C \to \mathbf{P}^1$ is a connected finite etale Galois cover of degree $n$. We must show $n=1$.
The sheaf $E := ...
17
votes
Accepted
Example of a smooth projective family of varieties in characteristic $p$ where the Hodge numbers jump
Update: The details of this construction are now available in my blog post with Sean Cotner on Thuses.
I was recently interested in exactly the same question. But I failed to find any reference where ...
15
votes
Accepted
Lang's Jacobian identity: slicker, elementary proof?
Awesome question! I haven't looked at Lang's paper yet, so I can't comment on whether this will be a different approach, but it is elementary. I will make use of Glynn's determinant formula at some ...
12
votes
Accepted
Does perfect fraction field imply perfect residue field?
The answer is no. Let $F$ be any imperfect subfield of a perfect field $F'$. Let $B$ be the ring of integral Puiseux series over $F'$ (integral meaning ones involving only nonnegative powers of $T$) ...
11
votes
Accepted
p-torsion in the Picard group of a regular projective curve
Take $p=3$ and $C\subset \mathbb{P}^2$ with equation $y^2 z=x^3 - t z^3$ where $t\in K$ is not a cube. Then $C$ is regular but, putting $L:=K(t^{1/3})$, $C_L$ is isomorphic to the usual cuspidal cubic ...
8
votes
Accepted
Residues of $\frac{1}{\prod_{i=1}^n (x-P_i)^{e_i}}$
If we restrict to the case when $e_1=e_2=\cdots=e_n=2$ the right condition is $\sum_{i=1}^n e_i=n+kp$ or $n+kp+1$ for some $k\geq 1$. This provides counterexamples to the stated conjecture (see below) ...
8
votes
Accepted
Hochschild Kostant Rosenberg theorem for varieties in positive characteristic?
See this paper of mine and Gabriele Vezzosi. We prove that HKR holds in particular for smooth proper schemes $X$ of dimension at most $p$, the characteristic prime. In particular, it holds for smooth ...
8
votes
A short proof for simple connectedness of the projective line
You can apply the following statement to $X = \mathbb{P}^1_K$ and $L = O(1)$ when $K$ is a separably closed field.
Let $L$ be a line bundle on a reduced connected scheme $X$ such that $H^{0}(X,...
8
votes
Accepted
Relation between positive roots of $E_8$ and $\mathbb{F}_2^8 \setminus \{0\}$
Let $L$ denote the root lattice of $E_8$. The group $W(E_8)$ acts (linearly) on $L/2L \cong \mathbb{F}_2^8$, and hence as a permutation of $2^8=256$ elements. This permutation action has the following ...
8
votes
Chromatic representation theory of the symmetric groups?
The coefficient ring of $K(h)[\Sigma_n]$ is, in degree zero, $\Bbb F_p[\Sigma_n]$, the group algebra on $\Sigma_n$ over $\Bbb F_p$. As a result, the list of idempotents in this ring is the same as for ...
7
votes
Lefschetz fixed-point theorem for the Frobenius map
For endomorphisms of elliptic curves, the Lefschetz fixed-point theorem goes back to Hasse and Deuring in the 1930s. See Silverman III 8.6 for the proof. For abelian varieties, it was proved by Weil ...
7
votes
Accepted
A variant on characteristic $p$ de Rham cohomology
To be able to compute the iterates of the Cartier operator it is convenient to understand how $C$ interacts with the de Rham differential:
Cartier isomorphism induces an isomorphism of complexes $$(\...
7
votes
Explicit large finite fields in characteristic $2$
The following paper provides an explicit computational approach to your problem:
J. D. Swift: Construction of Galois fields of characteristic two and irreducible polynomials, Math. Comput. 14, 99-103 (...
6
votes
Equivalent statements of the Riemann hypothesis in the Weil conjectures
Regarding the question about higher dimensions, Scholl showed that RH in all dimensions follows from the statement:
If $X$ is a smooth hypersurface in $\mathbb{P}^{d+1}$ over $\mathbb{F}_q$, then $X(\...
6
votes
Example of a smooth projective family of varieties in characteristic $p$ where the Hodge numbers jump
This is a comment on gdb's gorgeous answer, for people like me who aren't so comfortable with the Picard functor. Let $X$ be a space with a free action by one of the groups schemes $\mathbb{Z}/p$, $\...
Community wiki
6
votes
Accepted
Degree-2 étale covers of curves in characteristic 2 vs torsion points on the Jacobian
There is a duality between degree 2 coverings and two-torsion points on the Jacobian — i.e. both form elementary abelian 2-groups, and these groups are naturally dual.
This is the Artin–Milne Poincaré ...
6
votes
Accepted
Derived subalgebra of a restricted Lie algebra
There is no such a counterexample, as the derived subalgebra of the restricted Lie algebra associated to an associative algebra over a field of positive characteristic is always restricted.
For ...
5
votes
Accepted
Restriction of separable map
Let $k$ be an algebraically closed field of characteristic $2$. Consider $$Y=\text{Spec}\ k[t,v,w]/\langle w^2+v^3+tv^2\rangle, \ \ X=\text{Spec}\ k[t,u].$$ Let $f$ be the morphism determined by the ...
Community wiki
5
votes
Accepted
Harish-Chandra isomorphism for characteristic $p$
In the previous lines it is proven that $U(T)^W$ is integral over $\gamma(Z^{\mathcal G})$. Since $U(T)^W\subset Frac(U(T)^W)=Frac(\gamma(Z^{\mathcal G}))$ the equality follows from $\gamma(Z^{\...
5
votes
Singularities of curves that are moving
I am just writing an answer summarizing the counterexamples from the comments and adding one positive result for small degree.
Let $k$ be an algebraically closed field. Let $X$ and $Y$ be quasi-...
Community wiki
5
votes
Accepted
Do regular (but non-smooth) conics over a discretely valued field of characteristic $2$ admit a regular model over the valuation ring?
In general this isn't possible. The material that follows is a bit technical, but I do not have the time to explain all the details here now.
Suppose that $a, b \in T$ and that $a, b$ become squares ...
5
votes
Accepted
Do Neron-Severi groups of smooth projective unirational varieties contain $\ell$-torsion?
It seems that existence of $\ell$-torsion is possible. Here's one example.
Let $k$ be an algebraically closed field of characteristic $p$. In the paper "An Example of Unirational Surfaces in ...
5
votes
Explicit large finite fields in characteristic $2$
Nice question!
The case of self-reciprocal irreducible trinomials and pentanomials is of interest in cryptography.
The linked paper which I was not aware of before your question has the following ...
5
votes
Accepted
Explicit large finite fields in characteristic $2$
For $n=3^k$, the polynomial $p=x^{2n}+x^n+1$ is irreducible over $\mathbb{F}_2$.
Proof: By Rabin's irreducibilty test, it suffices to check that $p|x^{2^{2n}}-x$ and $\gcd(p,x^{2^{2n/3}}-x)=1$.
Note ...
5
votes
Accepted
Do we have Hodge symmetry for char $p$?
Hodge symmetry fails in positive characteristic in general; see Serre's Mexico paper [Ser58, Prop. 16] (for a more modern/conceptual version of that argument, see e.g. [vDdB21, Prop. 1.4]).
However, ...
5
votes
Accepted
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 $\overline{k(X)}$
Yes, if a specialisation of a polynomial has two distinct roots, then the original polynomial has two distinct roots, in the sense that you discuss.
Without loss of generality, $k$ is algebraically ...
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