All Questions
1,159 questions
4
votes
1
answer
369
views
Gauss mapping in finite characteristic
Suppose that $X\subset\mathbb P^n$ is a $d$-dimentional smooth projective variety (not a linear subspace) over an algebraically closed field. If $\gamma\colon X\to\mathrm{Gr}(d,\mathbb P^n)$ is Gauss ...
- 1,939
4
votes
1
answer
918
views
Heisenberg group in characteristic two
I have seen two constructions called the Heisenberg group. If $k$ is a field of characteristic not equal to $2$ and $V$ is a $2n$-dimensional vector space over $k$ with symplectic form $\omega : V \...
- 3,605
4
votes
1
answer
500
views
technical question about a paper on Belyi's theorem
I'm trying to understand Belyi's theorem, as presented here: http://eprints.soton.ac.uk/29785/1/b45h1koe.pdf
He defines a curve $X$ over a field $C$ to be a smooth projective geometrically connected ...
- 10.7k
4
votes
1
answer
674
views
Inseparable Galois Cohomology
First let me give a general form of my question, and then I'll give some motivation and a more specific version of it. Let $K/k$ be a Galois extension of fields with Galois group $G$, and let $X$ be ...
- 1,199
4
votes
2
answers
694
views
Ample line bundle and Frobenius morphism on smooth toric variety
Let $k$ be an algebraically closed field of $\mathrm{char}(k)=p>0$, $X$ a smooth toric projective variety of $\dim X=n$, $F_X:X\rightarrow X$ the absolute Frobenius morphism of $X$. Then for any $\...
- 85
4
votes
1
answer
457
views
If $f:X\to S$ is a universal homeomorphism, is $f':X\times_S X\to X$ a nil-immersion?
If $f:X\to S$ is a universal homeomorphism, is $f':X\times_S X\to X$ always a nil-immersion? This seems to be easy, yet possibly I miss something. Should I give references to this fact in a paper?
- 16.9k
4
votes
2
answers
402
views
lower bound for torsion of abelian varieties
Let $A$ be an abelian variety defined over a field $K$ of characteristic $p>0$. Let $A[\ell]$ be the group of $\ell$-torsion points, $\ell\neq p$ a prime. Are there positive constants $C, \eta$ ...
4
votes
1
answer
219
views
Does a smooth relative curve $X/S$ embed into $\mathbb{P}^3_S$?
Suppose that $\pi:X\to S$ is a smooth projective morphism of relative dimension 1. If $S$ is the spectrum of an algebraically closed field, then it is known that $X$ embeds into $\mathbb{P}^3_S$.
...
- 1,142
4
votes
1
answer
279
views
Is quotient of projective scheme over arbitrary base by a finite group also projective
This question probably follows from standard geometric invariant theory. If true I'd to know a reference for it.
Given a projective scheme $X\rightarrow S$ over the base $S$. Let's assume a finite ...
- 5,901
4
votes
1
answer
980
views
Geometric interpretation of sections $H^0(\Theta_X, X)$ of the Tangent sheaf over curve
I'm reading Mumford's & Oda's Algebraic Geometry II and I'm confused about explanations on geometric intuition of sections $H^0(\Theta_X, X)$ of the tangent sheaf on page 287:
Let $X$ a ...
- 6,016
4
votes
1
answer
557
views
Is a universally closed monomorphism a closed immersion?
The question is essentially in the title: $f\colon X \rightarrow Y$ is a monomorphism of schemes that is universally closed; does this imply that $f$ is a closed immersion? Any such $f$ is quasi-...
- 2,663
4
votes
1
answer
804
views
Pushouts of schemes along closed immersions
Is there a closed immersion $i:Z\to X$ and a finite morphism $f:Z\to Y$ of schemes such that the pushout of the span $Y\stackrel{f}{\leftarrow} Z\stackrel{i}{\rightarrow} X$ does not exist in the ...
- 349
4
votes
1
answer
272
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?
...
- 41
4
votes
1
answer
235
views
Which valuations of a field yield codimension $1$ subschemes of their 'models'
For a field $F$ (for example, a one generated by a finite number of its elements) there is a directed set of its 'models' (in this case those are 'arithmetic' schemes whose fraction field is $F$). It ...
- 16.9k
4
votes
1
answer
796
views
Deformation space of non-ordinary abelian varieties
It is a well known result of Serre and Tate that if $A$ is an ordinary abelian variety over a field $k$ of characteristic $p>0$, then the deformation space $\mathcal{M}$ of $A$ to an abelian ...
- 395
4
votes
1
answer
227
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 ...
- 309
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
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
4
votes
1
answer
156
views
Group algebraic spaces that are locally of finite type and have only finitely many points
Let $k$ be an algebraically closed field of characteristic zero. Let $G$ be a group algebraic space over $k$ such that $G\to $ Spec $k$ is locally of finite type.
Suppose that $G(k)$ is finite.
...
- 181
4
votes
1
answer
622
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{...
user19475
4
votes
1
answer
309
views
infinitesimal lifting criterion for non-noetherian schemes
We have the "standard criterion" which says that a morphism $f:X\rightarrow Y$ is smooth if:
1/ $Y$ is locally noetherian.
2/ $f$ is locally of finite type and satisfies lifting criterion for ...
- 3,472
4
votes
1
answer
301
views
Stacks with a small coarse moduli space
Let $k$ be a field of characteristic zero.
Let $X$ be a finite type algebraic stack over $k$ with a coarse (or good) moduli space $M$.
Suppose that $M$ is isomorphic to a point, i.e., $M = Spec k$.
...
- 41
4
votes
1
answer
827
views
When singular points of a reduced scheme are not dense in it?
A stupid AG question: could singular (Zarisky) points be dense in a reduced (Noetherian) scheme $S$? If yes, which 'standard' restrictions on $S$ could ensure that this does not happen? For example, ...
- 16.9k
4
votes
1
answer
322
views
Pairing on arithmetic surfaces
Let $f: X \to S$ be an arithmetic surface, where $S=\operatorname{Spec } O_K$ for a number field $K$. It is well known that if we want to introduce a reasonable intersection theory on $X$ we have to ...
- 321
4
votes
1
answer
861
views
Properties of log smooth schemes
Let $k$ be a field and $M$ be sharp monoid (with no invertible element) consider the log point $\eta_M=(\operatorname{Spec}(k), M)$. Let $X$ be a fine saturated scheme over $\eta_M$ such that the ...
- 171
4
votes
2
answers
1k
views
Is the absolute Galois group of the field of Laurent series in positive characteristic finitely generated?
If $K$ is an algebraically closed field of characteristic $p>0$, then $K((t))$, the field of Laurent series with coefficients in $K$, has infinitely many Galois extensions of degree $p$. Indeed, ...
- 3,362
4
votes
1
answer
221
views
Do permutation modules of solvable groups have self-dual socle in characteristic 2?
I was searching through the small groups database in GAP to find counterexamples to a certain conjecture (which is not important here). I was checking non-nilpotent solvable groups and noticed that ...
- 143
4
votes
1
answer
446
views
Exact functor in syntomic cohomology
By Tag 04C4 of the Stacks Project, for $f:X\rightarrow Y$ a closed immersion of schemes, the pushforward $f_*$ is exact for abelian sheaves on the big syntomic site.
Is it also true for a finite flat ...
- 3,472
4
votes
1
answer
360
views
Construct morphisms of schemes on level of associated functors
I have a general question about techniques used in @Emerton's proof, sketched below, in the answer to $\mathbb{P}^n$ is simply connected.
Given a finite étale map $\pi: Y \to \mathbb P^n$ (we regard ...
- 6,016
4
votes
1
answer
334
views
Affine open with irreducible complement
Let $X$ be an integral Noetherian separated scheme. Under what conditions can we find a non-empty affine open in $X$ whose complement is irreducible?
user139859
4
votes
1
answer
550
views
Dualizing sheaf and determinant of cohomology
Let $\pi:X\to S=\operatorname{Spec } O_K$ be an arithmetic surface in the sense of Arakelov geometry. Here $K$ is a number field $\pi$ is a flat map and $X$ is a projective surface. For any coherent ...
- 141
4
votes
1
answer
637
views
What sort of ind-scheme is this?
It apparently follows from work of Velu (MathSciNet) that every isogeny between elliptic curves in (long) Weierstrass form over $k$ can be written in the form
$$
\left(\frac{u(x)}{v(x)}, \frac{s_1(x)+...
- 35.5k
4
votes
1
answer
890
views
The SGA1 version of Riemann's Existence Theorem is about analytic spaces. How does one relate it to topological covering spaces?
In SGA1, Theoreme 5.1 (Riemann's Existence Theorem) states:
Let $X$ be a $\mathbb{C}$-scheme locally of finite type, $X^{\operatorname{an}}$ the associated analytical space. Then the functor which ...
- 49
4
votes
1
answer
628
views
Can the projection (tensor algebra) -> (symmetric algebra) be forced to split in char. p by factoring out p-th powers?
Question 1 (the weak and simple statement, which, I think, already is wrong): Let $p$ be a prime. Let $k$ be a field with characteristic $p$.
For any $k$-vector space $V$, consider the canonical ...
- 33.8k
4
votes
0
answers
178
views
Splitting of counit-trace map for $\ell$-adic sheaf $\Bbb Q_{\ell}$
I have a question about following argument on page 3 in paper arxiv.org/abs/1702.04404 by Will Sawin (Proposition 3):
The claim is that for a finite, dominant map $f:X \to Y$ between varieties $X,Y$ ...
- 6,016
4
votes
0
answers
200
views
Questions about the fixed point functor $X^G$ of a $G$-scheme
Let $X$ a (locally Noetherian; but not sure if that's really matter) $k$-scheme, $G$ a $k$-group scheme acting on $X$ via morphism $a:X \times G \to X$.
The fixed point functor of $X$ (where $X$ is ...
- 6,016
4
votes
0
answers
135
views
Specialization map Chow groups preserves algebraic equivalence
Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$.
Let $\pi\colon X\rightarrow \text{Spec}(R)$ be a smooth projective morphism with geometrically integral fibers.
In ...
- 984
4
votes
0
answers
108
views
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?
- 679
4
votes
0
answers
135
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 ...
- 2,751
4
votes
0
answers
183
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 ...
- 41
4
votes
0
answers
396
views
Non-Noetherian (classical) algebraic geometry
My starting point for this question is that, in a very classical sense, algebraic geometry is the study of solution spaces of systems of polynomial equations over an algebraically closed field. It is ...
- 365
4
votes
0
answers
286
views
Dévissage for a stratification in Grothendieck's Esquisse d’un programme: What is it?
I have a question about the the intuition of what Grothendieck proposed as tame topology in his "Esquisse d’un programme" as a "better suited" geometric structure in order to have ...
- 6,016
4
votes
0
answers
184
views
smooth super scheme which is not smooth
I am following the very nice "Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes" by Bruzzo, Ruiperez and Polishchuk. I am having some problem in order to give ...
- 558
4
votes
0
answers
103
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 $...
4
votes
0
answers
77
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 ...
- 13k
4
votes
0
answers
64
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 ...
4
votes
0
answers
419
views
Henselization of normal rings (Milne's EC)
The usual way to define the Henselization $A^h$ of a local ring $(A, \mathfrak{m})$ is by taking the direct limit $\varinjlim (B, \mathfrak q)$ over all étale neighborhoods of $A$
(i.e. pairs $(B,\...
- 6,016
4
votes
0
answers
284
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 $...
- 685
4
votes
0
answers
147
views
Uniqueness of Galois descent
Let $Y,Z$ be $\mathbb{F}_p$-schemes, they are both models of scheme $X$ over $\overline{\mathbb{F}_p}$ . Let $F$ be the absolute Frobenius of $\overline{\mathbb{F}_p}$, If $1_Y\times F$ and $1_Z\times ...
- 653
4
votes
0
answers
296
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 $\...
- 218