All Questions
1,159 questions
4
votes
0
answers
204
views
Explicit description of wonderful compactification for PGL_3
Let $k$ be an algebraically closed field of positive characteristics. Let $X$ be the wonderful compactification of $PGL_3$ (see for example section 6 of "Frobenius Splitting Methods in Geometry ...
- 163
4
votes
0
answers
317
views
Is the "naive" version of Chevalley's theorem still true?
Reposting from math.se in case more people are interested here.
Chevalley's theorem says that if $f \colon X \to Y$ is a morphism of finite presentation of schemes and $C \subset X$ is constructible, ...
- 584
4
votes
0
answers
215
views
Reference request: Radicial morphisms & Jacobson-Bourbaki correspondence
My name is Chemy (Przemysław). I am a PhD student at UvA (Amsterdam), and I work on projects related to foliations in algebraic geometry in positive characteristic. Therefore I am avidely reading two ...
- 485
4
votes
0
answers
231
views
How big are small inverse powers of 2 mod powers of 3?
Let $a \bmod b$ take value as an integer in $[0, b)$. For any $T \ge 1$, for what $R \in [0, 3^n)$ is
$$\min \{2^{-t}\bmod 3^n: t =1, \dotsc, T\} := \min A_T > R?$$
When $T$ is fixed as $n$ ...
- 459
4
votes
0
answers
189
views
If the Frobenius endomorphism of a characteristic $p$ ring is epimorphic, is it surjective?
MO question 19282 is about integral epimorphisms of commutative rings, and a counterexample is given to surjectivity. What about the case of the Frobenius endomorphism of a commutative, characteristic ...
- 4,492
4
votes
0
answers
262
views
de Rham Bloch-Ogus theory in positive characteristic
In their famous paper Gersten's conjecture and the homology of schemes, one of the results that Bloch and Ogus prove is that the second page of the coniveau spectral sequence for $X$ smooth over a ...
- 2,054
4
votes
0
answers
319
views
Is the restriction of an injective sheaf on a closed subscheme still injective?
Let $X$ be a Noetherian scheme, and let $i:Z\to X$ be the inclusion of a closed subscheme $Z$. Let $\mathcal{I}$ be an injective sheaf of modules on $X$.
Question. Is $i^*\mathcal{I}$ still an ...
- 1,479
4
votes
0
answers
130
views
Castelnuovo–Mumford regularity and wedge powers in positive characterisitc
A vector bundle on $\mathbb{P}^n$ is said to be $r$-regular if
$$H^i(\mathbb{P}^n,F(r-i))=0$$ for all $i>0$.
It is always true that if $F$ is $r$-regular and $G$ is $s$-regular (both vector bundles)...
- 2,356
4
votes
0
answers
233
views
Gluing two affine schemes along a different intersection
Given a quasi-projective $X$ variety that is the union of two affines $\text{Spec(A)}$, $\text{Spec(B)}$ with intersection $\text{Spec}(C)$. Let $f\in C$. Then $\text{Spec}(C_f)$ is an open in both of ...
- 5,901
4
votes
0
answers
169
views
Extensions of fraction field and residue field
Let $A \subset B$ be integrally closed local domains, $K(A), K(B)$ be fields of fractions, and $k(A),k(B)$ be residue fields. How to prove $[K(B):K(A)] \ge [k(B):k(A)]$?
This question should be easy ...
- 393
4
votes
0
answers
286
views
An application of Grothendieck's version of Hensel's Lemma
Assume $R$ is a Henselian local ring with $k=R/m_R$ ($m_R$ is the unique maximal ideal of $R$) and $G$ a finite flat group scheme over $R$. We denote by $G_k= G \otimes_R k$ the closed fiber.
There ...
- 6,016
4
votes
0
answers
133
views
Non-cyclic Galois groups over the field of formal Laurent series in positive characteristic
This should be an easy question, but I am unfortunately not able to give an answer, so I am sorry if this is not the appropriate level for the site.
Let $C$ be an algebraically closed field of ...
- 41
4
votes
0
answers
96
views
Finding a cover of the Hilbert functor, that corresponds to a cover of the Grassmannian
Let $\mathrm{Grass}_{k+1,n+1}:\mathrm{Sch}^{\circ}\rightarrow\mathrm{Set}$ be the Grassmann functor, which maps a scheme $S$ to the set:
$$\left\{\mathscr{U}\subseteq\mathscr{O}_S^{n+1}:\,\,\mathscr{...
- 215
4
votes
0
answers
169
views
Fibered surfaces degenerating to Frobenius
Let $R$ be a DVR with algebraically closed, positive characteristic residue field $k$. Let $X\rightarrow Spec(R)$ and $C\rightarrow Spec(R)$ be smooth projective morphisms of relative dimension 2 and ...
- 599
4
votes
0
answers
729
views
Why are algebraic schemes called algebraic?
In scheme theory, an algebraic scheme is the data of a scheme + a morphism of finite type to the spectrum of a field. Where does the term "algebraic scheme" come from? It does not seem intuitive to me ...
user141498
4
votes
0
answers
162
views
Embed FPPF group scheme into smooth one
Let $A$ be a ring and $G$ be an affine commutative FPPF group scheme over $A$. Can we embed $G$ into a smooth group scheme over $A$?
- 3,472
4
votes
0
answers
235
views
Krull dimension of schemes locally of finite type over PID
Let $R$ be a commutative unital ring that is a PID. Assume that $R$ is not a DVR. Let $X$ be an integral scheme locally of finite type over $\mathrm{Spec}\,R$. Can the Krull dimension of $\mathcal{O}...
user138661
4
votes
0
answers
483
views
A slightly canonical way to associate a scheme to a Noetherian spectral space
Let $C$ be the category whose objects are Noetherian spectral topological spaces and whose morphisms are homeomorphisms. Let $\mathrm{AffSch}$ be the category of Noetherian affine schemes (morphisms ...
user139951
4
votes
0
answers
275
views
Symmetric power contained in tensor power?
Let $V$ be an $R$-module. Traditionally the symmetric algebra $S(V)$ is defined as a quotient of the tensor algebra $T(V)$, by the ideal generated by all $a\otimes b-b\otimes a$.
Can $S^n(V)$ also be ...
- 2,519
4
votes
0
answers
468
views
Quaternion algebras in characteristic 2
Let $k$ be a field and let $Q$ be a quaternion algebra over $k$.
It is well known that, if $\mathrm{char}\,k\neq 2$, one can define $Q$ as the $k$-algebra of dimension $4$ generated by elements $x,y$ ...
- 375
4
votes
0
answers
834
views
When spreading out a scheme, does the choice of max ideal matter?
I'm looking at Serre's paper How to use Finite Fields for Problems Concerning Infinite Fields. Specifically I'm trying to use the techniques in the proof of Theorem 1.2 to write out the details of the ...
- 1,161
4
votes
0
answers
175
views
On representable abelian sheaves vs abelian sheaves
Let $S$ be a scheme, $(\text{Sch}/S)_{\rm Ét}$ a big étale site, and $A$ a representable
(either in schemes or algebraic spaces over $S$) abelian sheaf on $(\text{Sch}/S)_{\rm Ét}$.
Suppose there is a ...
user92332
4
votes
0
answers
537
views
When is a coherent subsheaf determined by its global sections
I am reading an article in which a proof is based on defining a subsheaf by only giving its global sections.
The exact setting is that, one has a surjective finite morphism $f:Y\to X$ between ...
- 333
4
votes
0
answers
206
views
A suspected typo, and Deligne's image of the general fiber swallowing the special
In SGA 4.5 (Arcata V.1) Deligne writes:
Let $X$ be a complex analytic variety and $f:X\rightarrow D$ map $X$ into the
disk. Write $[0,t]$ for closed line segment with extremities 0 and $t$ in
...
- 11.2k
4
votes
0
answers
169
views
Quotients of quasi affine varieties and extension of scalars
I have some questions about GIT quotients and extensions of scalars of categorical quotients:
1) Let $X$
be a complex algebraic quasi-affine variety, $G$
an algebraic reductive group over $\...
4
votes
0
answers
135
views
henselizations along closed subscheme
Where can I find some references about henselizations ablong a closed subscheme?
For example if I take a map $Y\times\mathbb{A}^{1}\rightarrow Y$ and $Z$ a closed subscheme.
Let $Y_{Z}^{h}$ the ...
- 3,472
4
votes
0
answers
285
views
Application of Frobenius splitting in characteristic $0$
In general, Frobenius splitting only defines on field of characteristic $p$ (algebraically closed) field.
I am reading Brion and Kumar's book and I can see that there are geometric results can be ...
- 849
4
votes
0
answers
378
views
Properties of schemes determined by field valued points [closed]
Are there any interesting cases (interesting here is interpreted rather loosely here) where you can show $X$ has property $P$ whenever all $X(K)$ have property $P$ where $K$ runs through all fields?
...
- 3,968
4
votes
0
answers
209
views
Partial differential Equation over characteristic p
I want some references on partial differential equations over characteristic $p$. If we have a first order partial differential equation, how can we check whether there exists polynomials or rational ...
4
votes
0
answers
326
views
Tannaka categories and reductive groups
The group associated to a Tannaka category $T$ over a field is pro-reductive if and only if $T$ is semi-simple.
Pro-reductive groups make sense over any scheme.
Is there an extension of the theory ...
- 71
4
votes
0
answers
185
views
Are these subspaces of $\mathbb{Z}/3[[x]]$ stable under the shallow Hecke algebra?
This is a characteristic $3$ analog of part of my earlier question, "Are these two subspaces of $\mathbb{Z}/2[[x]]$ the same?"
Notation
Fix a prime $N$ other than $3$. Let $F,G \in \mathbb{Z}/3[[x]]$...
- 5,422
4
votes
0
answers
164
views
Is there an analogue of distributions in characteristic p?
Some motivation: When working over $\mathbb{C}$, distributions (in the sense of generalized functions) act as natural generators for $D$-modules (in the sense that any regular holonomic $D$-module is ...
- 1,488
4
votes
0
answers
189
views
Does this space of mod 2 modular forms admit a (Z/8)* degree decomposition?
Fix an odd N>0. Let M consist of all odd elements of Z/2[[x]] that are the mod 2 reductions of elements of Z[[x]] arising as the Fourier expansions of modular forms for (Gamma_0)(N); it's easy to see ...
- 5,422
4
votes
0
answers
814
views
Adjunction Formula for Weil Divisors on a Normal Variety X
Let $X$ be a normal variety over an algebraically closed field $k$ of characteristic $p>0$ and $S$ be a prime Weil divisor on $X$ which is normal too. Now if $K_X+S$ is NOT $\mathbb{Q}$-Cartier, ...
- 165
4
votes
0
answers
136
views
A subring of the Serre Swinnerton -Dyer ring of level N modular power series
Suppose ell is prime and (N,ell)=1. Consider those power series over Z that are expansions at infinity of modular forms for gamma_0 (N) of weight a multiple of ell-1. I'll say that an element of (Z/...
- 5,422
4
votes
0
answers
395
views
G-torsor whose ring of regular functions is a field.
I already asked this question on stackexchange but didn't get any answer. Maybe it is better suited for mathoverflow.
Let $G$ be an affine group scheme (not necessarily of finite type) over $\...
- 7,527
4
votes
0
answers
443
views
What is known about "singularity types" in the Murphy's Law sense?
In his "Murphy's Law" paper, Vakil gives a definition equivalent to the following:
The singularity type of a pointed scheme $(X,p)$ its equivalence class, under the following equivalence relation: $...
- 7,338
4
votes
0
answers
197
views
Inequalities between numerical invariants of nonsingular projective Varieties in positive Characteristic
It is well-known that Miyaoka and Yau-type inequalities do not hold in positive characteristic. In "a note on Bogomolov-Gieseker’s inequality in positive characteristic", however, we can ...
- 4,902
4
votes
0
answers
987
views
Quotient morphisms in the category of schemes
Which morphisms of schemes (or varieties, if you prefer) $\pi: X \rightarrow Y$ are quotient morphisms, i.e. satisfy the following universal property (*)?
(*) For any morphism $f:X \rightarrow Z$, ...
- 23.4k
3
votes
2
answers
2k
views
Smooth morphism (algebraic geometry) vs. Submersion (differential geo) & Ehresman's Lemma
I have a general question about the motivation behind to definition the smooth morphisms
as we know it from algebraic geometry. The most common
definition of a smooth morphism $: X \to Y$ between two ...
- 6,016
3
votes
2
answers
635
views
Push-forward of a quasi-coherent graded algebra under a proper map
Let $f\colon X \rightarrow Y$ be a proper morphism with $Y$ Noetherian (and even affine, if you wish), and let $\mathscr{A} = \bigoplus_{n \ge 0} \mathscr{A}_n$ be a quasi-coherent graded $\mathscr{...
- 1,103
3
votes
2
answers
613
views
tamely branched cover over P^1
k is an algebraically closed field, X is a smooth, connected, projective curve over k. f: X-->P^1 is a finite morphism. Let t be a parameter of P^1, suppose f is etale outside t=0 and t=\infty, and ...
- 1,091
3
votes
1
answer
397
views
Flat cover by a locally Noetherian scheme
Les S be a scheme. Does there exist a faithfully flat morphism T to S with T a locally Noetherian scheme?
- 137
3
votes
2
answers
390
views
The underlying space of an affine open dense subscheme
Let $X$ be a Noetherian scheme, $U\subset X$ be an affine open dense subscheme. Is the underlying space of $U$ necessarily homeomorphic to the underlying space of $X$?
user140689
3
votes
1
answer
546
views
What if the base change of an algebraic space is representable
Let $k\subset L$ be an extension of fields of characteristic zero.
Suppose that $X/k$ is an algebraic space such that $X\otimes_k L$ is representable by a finite type $L$-scheme.
I am sure there are ...
- 33
3
votes
2
answers
243
views
Minimal fields of isomorphism for varieties
Let $V$ be an algebraic variety over a field $K$. Is there a constant $d = d(V) \in \mathbb{N}$ such that for any variety $W$ defined over $K$ and isomorphic to $V$ over the algebraic closure of $K$, ...
- 11.3k
3
votes
2
answers
472
views
Induced maps of an automorphism of a curve on the tangent ot its jacobian and on its differential forms
Let $C$ be a smooth projective curve of genus $g$ over a field $k$ and $J$ be its jacobian (defined over $k$). Let $\sigma: C \rightarrow C$ be a $k$-automorphsm of $C$. This automorphism $\sigma$ ...
- 1,109
3
votes
1
answer
405
views
Krull dimension of the ring of global sections
Let $X$ be an irreducible scheme. Can the Krull dimension of $\mathcal{O}_X(X)$ exceed that of $X$?
3
votes
1
answer
349
views
Field extensions in Grothendieck rings
Let $k$ be a field, and consider the Grothendieck ring of $k$-varieties, $K_0(V_k)$. Let $K/k$ and $K'/k$ be field extensions of $k$. We view $\mathrm{Spec}(K)$ and $\mathrm{Spec}(K')$ as $k$-schemes ...
- 4,605
3
votes
1
answer
278
views
"Covering-flat" part in definition of morphism of sites
Let $(\mathcal{C},\mathcal{I})$ and $(\mathcal{D},\mathcal{J})$ be sites where $\mathcal{C}, \mathcal{D}$ are categories and $\mathcal{I}$ and $\mathcal{J}$ are Grothendieck topologies on $\mathcal{C}...
- 6,023