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, ...
pozio's user avatar
  • 599
29 votes
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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 ...
David E Speyer's user avatar
22 votes
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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 ...
Wojowu's user avatar
  • 27.4k
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 ...
Alex Kruckman's user avatar
18 votes
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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 := ...
user117273's user avatar
17 votes
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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 ...
gdb's user avatar
  • 2,851
15 votes
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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 ...
Gjergji Zaimi's user avatar
12 votes
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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 $T$) ...
Wojowu's user avatar
  • 27.4k
11 votes
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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 ...
Laurent Moret-Bailly's user avatar
8 votes
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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) ...
Gjergji Zaimi's user avatar
8 votes
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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 ...
Benjamin Antieau's user avatar
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,...
js21's user avatar
  • 7,199
8 votes
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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 ...
Theo Johnson-Freyd's user avatar
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 ...
Tyler Lawson's user avatar
  • 51.1k
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 ...
anon's user avatar
  • 71
7 votes
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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 $$(\...
SashaP's user avatar
  • 7,027
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 (...
Francesco Polizzi's user avatar
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(\...
David E Speyer's user avatar
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 $\mathbb{Z}/p$, $\...
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 ...
Will Sawin's user avatar
  • 135k
6 votes
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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 ...
Salvatore Siciliano's user avatar
5 votes
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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 ...
5 votes
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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^{\...
SashaP's user avatar
  • 7,027
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-...
5 votes
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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 ...
darx's user avatar
  • 66
5 votes
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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 ...
Maciek's user avatar
  • 351
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 ...
kodlu's user avatar
  • 10.1k
5 votes
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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 ...
Antoine Labelle's user avatar
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, ...
R. van Dobben de Bruyn's user avatar
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 ...
LSpice's user avatar
  • 11.3k

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