All Questions
1,159 questions
4
votes
1
answer
1k
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Vector space structure on the tangent bundle of a scheme and relation to the tangent sheaf
First a word of warning: I am not a trained algebraic geometer, so it is possible (likely) that these questions are inappropriate for MO, if so: my apologies.
Said this: As far as I understand the ...
- 1,939
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
2
votes
1
answer
728
views
The restriction of a global section which is not a zero divisor is still a non-zero divisor?
Let $X$ be a scheme. $U$ is an open subscheme of $X$. Assume $f$ is a global section on $X$ which is not a zero divisor, then the restriction of $f$ to $U$ is still a non-zero divisor?
If $X$ is ...
- 873
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
3
votes
2
answers
1k
views
Components of an exceptional divisor
Let $X$ be a projective variety and let $\tilde{X}$ be the blow-up of $X$ at a subscheme $Z$. Let $F$ be the exceptional divisor of $\tilde{X}$. I wonder:
What is the number of irreducible ...
- 83
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
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
46
votes
0
answers
2k
views
Mikhalkin's tropical schemes versus Durov's tropical schemes
In Mikhalkin's unfinished draft book on tropical geometry, (available here) (page 26) he defines a notion of tropical schemes. It seems to me that this definition is not just a wholesale adaptation of ...
- 6,229
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
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
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
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
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
6
votes
1
answer
551
views
Limits of reduced schemes question from Eisenbud and Harris
My question pertains to exercise II-16 in Eisenbud and Harris' "The geometry of Schemes". For an algebraically closed field $K$ the question is as follows:
Consider zero-dimensional subschemes $\...
- 485
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
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
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
6
votes
1
answer
895
views
When does Zariski closure commute with base change?
This should be an elementary question for anyone who knows SGA by heart (alas, not for me). It smells a lot like a descent problem. All schemes are supposed to be noetherian, and all morphisms to be ...
- 4,015
4
votes
2
answers
2k
views
Hartshorne's associated scheme for a variety
This question comes from Proposition 2.6 in Chapter 2 of Hartshorne's Algebraic Geometry. In my edition, that's on page 78.
For a variety $V$, Hartshorne defines the topological space $t(V)$ to ...
- 235
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
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
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
5
votes
0
answers
234
views
Modular reduction of exceptional complex reflection groups
I am interested in reducing reflection representations of complex reflection groups modulo a prime $p$. For the infinite family $G(m,r,n)$, it is straightforward to get "good reduction" provided that $...
- 10.7k
7
votes
1
answer
2k
views
Can proper-smooth base change be used to show that varieties cannot be lifted to characteristic zero?
Recall the following corollary to the proper and smooth base change theorems:
Let $\pi: X \to S$ be a proper, smooth morphism. Then the direct images $R^i \pi_* \mathcal{F}$ are locally constant ...
- 25.6k
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
12
votes
4
answers
2k
views
Finite subgroups of $PGL_2(K)$ in characteristic $p$
Let $K$ be a field of characteristic $p$. What are the finite subgroups of $PGL_2(K)$ whose orders are divisible by $p$? And if $G$ and $H$ are two such subgroups that are isomorphic, can one say when ...
- 1,199
2
votes
1
answer
690
views
Restricted universal enveloping algebra of Abelian p-Lie algebra
Question: Let $p$ be a prime. Let $k$ be a commutative ring such that $p=0$ in $k$.
Let $\mathfrak g$ be an abelian $p$-restricted Lie algebra over $k$. In other words, let $\mathfrak g$ be a $k$-...
- 33.8k
5
votes
2
answers
3k
views
morphisms of affine schemes question
So, in chapter 2, section 2 of Hartshorne, (prop 2.3), he describes how if $\varphi : A\rightarrow B$ is a homomorphism of rings, then you get a morphism of (affine schemes):
$\newcommand{\Spec}{\...
- 10.7k
13
votes
2
answers
1k
views
Is the Characteristic of a Field Detectable from the Topology of a Topological Vector Space?
Motivation
A topological vector space is a vector space over a (topological) field, K, that carries a topology such that addition and scalar multiplication are continuous maps, e.g., all normed vector ...
6
votes
1
answer
2k
views
k rational points and base change
This could be a tricky question but could help me to better understand these very interesting things.
Let $X$ be an algebraic variety over a field $k$ (in the sense of a k-scheme like in Qing Liu), $...
- 379
7
votes
1
answer
1k
views
Closed points of valuation scheme
In the excellent book "Algebraic Geometry 1" of Görtz & Wedhorn, in exercise 3.14, one is asked to show that in the spectrum of a valuation ring with infinitely many primes, the complement of the ...
- 1,128
13
votes
1
answer
651
views
Help wanted with Chebotarev condition in characteristic 2
Having promised a longtime collaborator that I would clear my plate to finish up some joint work of ours, I am swallowing my pride and tossing up the following technical point of function field ...
- 65.4k
15
votes
0
answers
779
views
Lifting varieties from char. $p$ to char. 0 after alterations
The question is related to this MO question:
Lifting varieties to characteristic zero.
Let $X$ be a projective smooth variety over $k$ alg. closed field of char. $p.$ Does there always exist an ...
- 4,265
4
votes
2
answers
604
views
Adem-Wu relations from Bullett-Macdonald identities
Question. Let $p$ be a prime. Let $q$ be a power of $p$. Let $P^0$, $P^1$, $P^2$, ... be elements of some associative $\mathbb F_q$-algebra $A$. (Here, $P^i$ does not mean $P$ to the $i$-th power; ...
- 33.8k
2
votes
0
answers
631
views
cohomology of projective limit of schemes
Hello,
Suppose that $X_i$ is a projective system of schemes and $F_i$ is a compatible
projective system of abelian sheaves on the $X_i$ (i.e. if $p_{ij} : X_i \to X_j$ is the
transition map, then we ...
- 2,842
7
votes
1
answer
588
views
Complex analytic space with no (positive dim.) subscheme ?
Is there an example of a complex analytic space $X$ that doesn't have any (not necessarily open or closed) positive dimensional subspace $Y$ which is analytically isomorphic to (the complex analytic ...
- 23.4k
17
votes
2
answers
1k
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What is an explicit example of a variety X which is finite over Spec F_p but which does not lift to a scheme Y which is finite and flat over Spec Z_p?
What is an explicit example of a variety X which is finite over Spec F_p but which does not lift to a scheme Y which is finite and flat over Spec Z_p?
- 10.5k
5
votes
1
answer
461
views
Given a branched cover with branch cycle description $(g_1,...,g_r)$, does $g_i$ generate some decomposition group?
Classically:
Let $a_1,...,a_r$ be points in $\mathbb{P}^1_{\mathbb{C}}$, and let $\alpha_1,...,\alpha_r$ be simple loops around the $a_i$, all counterclockwise, and none touching (so $\alpha_1...\...
- 5,929
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
2
votes
1
answer
1k
views
Is every regular (excellent) scheme separated?
Sorry for one more stupid AG question. I need schemes that are regular, excellent and separated. Are these three conditions independent?
- 16.9k
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
2
votes
1
answer
332
views
Ample bundle under Frobenius morphism
Let $k$ be an algebraically closed field of char($k$)=p>0, $X$ a smooth projective variety over $k$, $F:X\rightarrow X^{(1)}$ the relative Frobenius morphism. Let $E$ be an ample vector bundle on $X$. ...
- 85
5
votes
0
answers
323
views
Vector bundles of schemes and their topological realizations
Hi, there is a realization functor $R_\mathbb{R}$ from schemes of finite type over $\mathbb{R}$ to topological spaces and there is also a functor $R_\mathbb{C}$.
Does $R_\mathbb{R}$ send an ...
- 103
5
votes
1
answer
401
views
Topological space associated to a real or complex scheme
Hi, consider a scheme $X$ of finite type over $\mathbb{R}$ (or $\mathbb{C}$). In Hartshorne's appendix B on 'transcendental methods' it is shortly mentioned how to assign a reasonable topological ...
- 103
0
votes
0
answers
352
views
Liftability in positive characteristic
What clsses of algebraic varieties over field of positive characteristic can be lift to $W_2(k)$?
- 85
7
votes
2
answers
513
views
Tameness for the Galois closure of a map of curves
Say we are working over some $K=\overline{K}$, of characteristic $p>0$. Let $\phi: Y\rightarrow X$ be a nonconstant map of smooth projective curves. To this map we can associate a map $\psi: Z\...
- 801
0
votes
0
answers
524
views
DeRham cohomology
The Poincare lemma fails in positive characteristic, since pth powers vanish under differention. My question is : is there still some kind of resolution of the local system k by considering some ...
- 1
2
votes
0
answers
370
views
Can any radiciel morphism be presented as the composition of a universal homeomorphism with an immersion?
Let $f:X\to Y$ be a radiciel (=universally injective) morphism, where $X$ is a regular connected scheme. Can it be presented as the composition of a universal homeomorphism with an immersion? This ...
- 16.9k
234
votes
16
answers
57k
views
What elementary problems can you solve with schemes?
I'm a graduate student who's been learning about schemes this year from the usual sources (e.g. Hartshorne, Eisenbud-Harris, Ravi Vakil's notes). I'm looking for some examples of elementary self-...