All Questions
2,494 questions
6
votes
1
answer
804
views
Del pezzo surfaces in positive characteristic
For me a Del Pezzo surface $X$ over an algebraically closed field of characteristic $p$ is an algebraic surface where the anticanonical bundle $\omega^{-1}_X$ or $-K_X$ is ample. (I prefer the second ...
- 2,215
1
vote
1
answer
443
views
What kind of conditions we need to make morphisms of schemes quasi-projective?
What kind of conditions we need to make morphisms of schemes quasi-projective?
I am really interested in the following case:
If $f : X \to Y$ is an etale, of finite type and separated morphism of ...
- 945
21
votes
1
answer
2k
views
When does the relative differential $df=0$ imply that $f$ comes from the base?
Let $A \to B$ be a map of commutative rings, and $d : B \to I/I^2$ be
defined by $df = f\otimes 1 - 1\otimes f$, where $I$ is the kernel of
$B \otimes_A B \to B$, as in [Hartshorne II.8].
If $df=0$,...
- 27.9k
0
votes
0
answers
159
views
a question on the Poincar\'e bundle
Let $C$, a smooth curve. Let $J$ its Jacobian, consider the Poincar\'e bundle $\mathcal{P}$ on $J\times J$. Let $p: J\times J\rightarrow J$ the projection.
How can I compute the complex $R p_{*} \...
5
votes
1
answer
446
views
More questions involving characteristic 2 theta series identities
In my answer to my earlier question, "Existence of certain identities involving characteristic 2 thetas", I established some curious identities when the thetas have prime "level" congruent to 1 mod 4 ...
- 5,422
6
votes
2
answers
540
views
Are the closures of the tori in the decomposition of a torified variety toric varieties?
In "Torified varieties and their geometries over $\mathbb{F}_1$", J. L. Pena and O. Lorscheid define a torified variety as a variety $X$ over $\mathbb{Z}$ along with a family of locally closed ...
- 3,290
0
votes
0
answers
148
views
Is sum $(E_i, E_j)$ non-positive, with $E_i$'s the exceptional components of a desingularization
Let $Y$ be an integral normal 2-dimensional scheme and let $X\longrightarrow S$ be a flat projective morphism, where $S$ is a Dedekind scheme.
Let $f:X\longrightarrow Y$ be a minimal resolution of ...
2
votes
1
answer
304
views
Connected extensions of finite by connected algebraic groups
Let $1\to H\to E\to G\to 1$ be a short exact sequence of algebraic groups defined over an algebraically closed field $k$ of characteristic $p$. Suppose $H$ is a finite group, and $G$ and $E$ are ...
- 631
6
votes
0
answers
1k
views
a naive question about p-adic local monodromy theorem
The question is about whether one can view the p-adic local monodromy theorem as the quasi-unipotence of some monodromy operator.
it is known that the classical local monodromy theorem (i.e. for ...
- 313
12
votes
3
answers
1k
views
Sequences of Squares with all square differences
Background
The following question was first asked by Alex Rice, who was thinking about small subsets $A\subset [1,\ldots , N]$ with lots of square differences. Certainly for any set $A$ the maximum ...
- 3,625
11
votes
1
answer
615
views
Do Richardson varieties have rational singularities in arbitrary characteristic?
The title basically asks the question. I'll review the relevant terminology and explain what I have and haven't found in the literature.
Let $G$ be a reductive group. Let $v \leq w$ be elements of ...
- 156k
13
votes
1
answer
793
views
Arithmetic and moduli spaces of higher genus curves
Modular curves (as moduli of elliptic curves with level structure) play a key role in the study of the arithmetic of elliptic curves. The higher genus curves have very different arithmetic, but I ...
- 605
5
votes
0
answers
708
views
Formal groups in the supersingular reduction case
Let $E/\mathbb{Q}$ be an elliptic curve with potential good supersingular reduction at $p$. Thus, there is a finite extension $K/\mathbb{Q}_p$ such that $E/K$ has good supersingular reduction. Let us ...
- 1,694
11
votes
1
answer
1k
views
Which curves have stable Faltings height greater or equal to 1
Let $Y$ be a smooth projective connected curve of genus $g>0$ over $\overline{\mathbf{Q}}$. Let $h_{\textrm{Fal}}(Y)$ be the Faltings height of $Y$.
Question 1. Can one classify or describe the ...
- 9,497
8
votes
1
answer
331
views
If C is a fusion category over a field of nonzero characteristic and dim C = 0, is Z(C) ever fusion?
If $C$ is a fusion category and $\dim(C) \neq 0$ (the latter is automatic in characteristic zero, but not in nonzero characteristic), then the Drinfel'd center $Z(C)$ is fusion. More generally, if $C$...
- 28.1k
1
vote
0
answers
108
views
Why do subspaces of the space of Global holomorphic differentials of a function field correspond to its subfields
I'm asking this question as a follow up to the Felipe Voloch's answer to this question:
Subfields of a function field
which you can read it here:
Subfields of a function field
(I just didn't have ...
- 601
3
votes
0
answers
281
views
What would be a characteristic-$p$ analogue for $C^{\infty}$-fiber bundles?
I'd like to know a notion for a morphism between algebraic varieties in characteristic $p$ that plays the role of a $C^{\infty}$-fiber bundle. It should be, in particular, flat. I'm not assuming the ...
- 4,265
12
votes
9
answers
6k
views
Proofs of Mordell-Weil theorem
I would like to ask if there exist pedagogical expositions of the Mordell-Weil theorem (wikipedia). What parts of number theory (algebraic geometry) one should better learn first before starting to ...
- 14.3k
3
votes
0
answers
302
views
Small primes as stepping stones
It is well-known that in his celebrated proof of Fermat's Last Theorem, Wiles made a crucial use of the result of Langlands and Tunnell to deduce modularity of the Galois representation on the 3-...
- 786
4
votes
0
answers
393
views
Motivic interpretation of genus 2 Siegel forms induced by lifts of Maass and Skoruppa
Background: There are several known lifts from integral weight modular forms to Siegel forms of genus 2, among them the Saito-Kurokawa lift. Another lift construction that is important for ...
- 1,704
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
2
votes
1
answer
455
views
good references in moduli stack and stable reduction
Hello,everyone. I'm looking for some good references in moduli stack and stable reduction, so I ask here for some advice.
I knew the famous paper of Deligne-Mumford, but this paper is hard for me ...
- 1,921
13
votes
1
answer
690
views
Obstructions to formally integrating vector fields in characteristic p?
Let $M$ be a smooth scheme over some field $k$ of characteristic $p$, and $\vec X$ a vector field on it. Equivalently, $\vec X$ gives a map $Spec\ k[\epsilon]/\langle \epsilon^2 \rangle \times M \to M$...
- 27.9k
5
votes
0
answers
518
views
Hochschild-Serre for hypercohomology
I need either a proof or a good reference for the following plausible statement:
Let $S$ be a scheme and let $C$ be a bounded complex of abelian sheaves on $S_{\rm{fppf}}$. Let $S^{\prime}\rightarrow ...
2
votes
2
answers
313
views
Is this morphism the normalization of P^1 in this curve
Let $S$ be an integral Dedekind scheme.
Let $f:X\longrightarrow \mathbf{P}^1_{S}$ be a finite flat surjective morphism, where $X$ is an integral normal scheme.
Let $\eta$ be the generic point of $S$...
- 23
12
votes
1
answer
2k
views
Replacement for derivations in characteristic p?
Let $k$ be a field.
If $f \in k[x]$ is a polynomial, and $d/dx\ f = 0$, then either
$f$ is constant, or
$char\ k = p$ and $f \in k[x^p]$.
So "annihilated by all derivations" is perhaps not the right ...
- 27.9k
0
votes
1
answer
414
views
Is the following morphism etale
Let $Y$ be a reduced noetherian $1$-dimensional scheme such that the normalization morphism $f:X \longrightarrow Y$ is finite. Let $g:Y\longrightarrow Z$ be a finite flat morphism, where $Z$ is a ...
- 3
7
votes
1
answer
748
views
Hodge spectral sequence for algebraic stacks
In a beautiful paper Deligne and Illusie have shown the following: Let $f\colon X \to S$ be a smooth proper morphism of schemes in characteristic $p > 0$, let $F\colon X \to X^{(p)}$ be the ...
- 1,645
2
votes
1
answer
264
views
intersecting sections on the projective line
This question is about intersection theory on the easiest (arithmetic) surface over $\mathbf{Z}$: $\mathbf{P}^1_{\mathbf{Z}}$.
Suppose we are given two distinct $\mathbf{Q}$-rational points $b_1$ and ...
- 73
4
votes
1
answer
844
views
intersection number
I vaguely recall the following fact that I'd like to use in my research. It should be easy to see that this holds (if it does) but I can't seem to prove it.
Let $p:X\longrightarrow S$ be a (regular) ...
- 73
21
votes
4
answers
2k
views
Simplest example of jumping of cohomology of structure sheaf in smooth families?
Using Hodge theory (and the ill-defined Lefschetz principle), one can show that in characteristic 0, given a proper smooth family $X \rightarrow B$, the cohomology groups of the structure sheaf of the ...
- 3,857
8
votes
2
answers
3k
views
Ramification divisor associated to a cover of a regular scheme
Let $S$ be the spectrum of $\mathbf{Z}$ or the spectrum of an algebraically closed field. (Actually, one can take $S$ to be any noetherian integral regular scheme.)
Let $f:X\longrightarrow Y$ be a ...
- 83
4
votes
1
answer
501
views
A question about moduli spaces over $\mathbb{Z}$
I've seen, on several occasions, papers whose purpose it is to construct a moduli space over $\mathbb{Z}$ for a moduli problem for which a moduli space over $\mathbb{C}$ was already constructed. Let's ...
- 5,929
2
votes
1
answer
202
views
In Riemann Existence, what is the interpretation of the space of complex-geometric points?
I've been thinking recently about moduli spaces defined over $\mathbb{Z}$, and this led me to the following question:
Question
Riemann existence says that if we have a variety over $\mathbb{C}$, $X_{...
- 5,929
0
votes
0
answers
263
views
Computing the function field of a curve given as a subvariety of the Jacobian of its cover or merely the degree of the covering
I read following paragraph from:
G. Tamme, Teilkörper höheren Geschlechts eines algebraischen Funktionenkörpers, Arch. Math. 23 (1972), 257--259
Here $C$ is a curve of genus $\ge 2$ and $J$ is the ...
- 601
0
votes
1
answer
448
views
Bilinear system of Diophantine Equations
$\forall i,j \in$ $\{$$1, \cdots,n$$\},$ let $x_{i},y_{i}$ be unknowns and $n_{ij} \in \mathbb{Z}$ with $i \le j$ be the knowns.
Consider the following $\frac{n(n+1)}{2}$ with $n > 2$ ...
- 800
13
votes
1
answer
1k
views
Is there a "trianguline period ring", or is one expected?
Consider a finite-dimensional $\mathbf{Q}_p$-vector space $V$ and a continuous representation $\rho : G_{\mathbf{Q}_p} \to \mathrm{GL}(V)$. Fontaine introduced various $\mathbf{Q}_p$-algebras with $...
- 13.1k
4
votes
1
answer
627
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
12
votes
1
answer
1k
views
Ramification in p-division fields associated to elliptic curves with good ordinary reduction
Dear MO,
Let $p$ be a prime and let $E/\mathbb{Q}_p$ be an elliptic curve. Suppose that $E/\mathbb{Q}_p$ has good ordinary reduction at $p$. In his 1972 paper ``Propriétés galoisiennes des points d'...
- 1,694
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 ...
- 2,976
1
vote
0
answers
239
views
Torsion points on commutative $Z_p$-group schemes
Hi,
Let G be a smooth commutative $\mathbb{Z}_p$-group scheme of finite type and let $G_0$ be the $\mathbb{Q}_p$-fiber. We have an embedding $G(\mathbb{Z}_p)\subseteq G_0(\mathbb{Q}_p)$. My question ...
- 527
32
votes
10
answers
3k
views
Which 'well-known' algebraic geometric results do not hold in characteristic 2?
A smooth curve $X$ in $\mathbb{P}^n$ is strange if there is a point $p$ which lies on all the tangent lines of $X$.
Examples are $\mathbb{P}^1$ is strange and so is $y=x^2$ in characteristic $2$. ...
- 2,215
18
votes
1
answer
3k
views
Field with one element example?
$$\frac{1}{\mu(B)}\int_B \vert x \vert d\mu(x) = \frac{1}{p+1}$$
This formula holds for the unit ball in $\mathbb{Q_p}$. This formula also holds for
$\mathbb{R}$ when $p=1$. Should one expect $$\...
- 370
0
votes
1
answer
462
views
Vector bundles' conjectures [closed]
Hi,
I know the title sounds too much general.
Googling the question I have not found much material, so I decided to ask to experts.
I would like to know which are the most famous/important unsolved ...
- 51
8
votes
1
answer
812
views
The Galois representation of a p-divisible group is crystalline
Can someone explain (or give a reference) why the Galois representation attached to a p-divisible group over the ring of integers of a p-adic ring is Crystalline?
- 137
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
2
votes
1
answer
568
views
Can we construct rational functions with prescribed ramification on an algebraic curve over \Qbar
Let $C$ be a smooth projective connected curve of genus $g$ over $\bar{\mathbf{Q}}$. Fix a finite non-empty (Edit) set of closed points $S$ in $C$ and let $U$ be the complement of $S$ in $C$.
Q1. (...
- 9,497
5
votes
2
answers
556
views
Existence of certain identities involving characteristic 2 "thetas"
Let l=2m+1 be prime. In my previous MO question, "What are the polynomial relations between these characteristic 2 thetas?", I defined a subring of Z/2[[x]] as follows:
The subring, S, is generated ...
- 5,422
13
votes
2
answers
1k
views
Families of curves for which the Belyi degree can be easily bounded
I know (edit: three) families of smooth projective connected curves over $\bar{\mathbf{Q}}$ for which the Belyi degree is not hard to bound from above.
The modular curves $X(n)$. They are ...
- 9,497
7
votes
2
answers
1k
views
How does one compute induced representations for modular representations?
The set-up is this: Let $G$ be a finite group, and $H$ a subgroup. We are given an irreducible representation of $H, \rho: H\rightarrow GL_n(K)$ (I will notationally identify $\rho$ with its character)...
- 1,386