All Questions
624 questions
2
votes
1
answer
451
views
Tie-Breaking Trick for Log Canonical Pairs and F-pure pairs in Positive Characteristic
Let $X$ be a projective 3-fold in characteristic $p>0$. Let $(X, D)$ be a klt pair, and $D'$ a $\mathbb{R}$-Cartier divisor such that $D'=A'+B'$, where $A\geq 0$ is an ample $\mathbb{Q}$-divisor ...
- 313
1
vote
0
answers
246
views
Frobenius twist of a field
Let $k$ be a field of characteristic $p>0$ (not necessarily perfect). Consider the Frobenius endomorphism $F : k \to k$, $x \mapsto x^p$. I am curious about what happens when we take $k$ as a $k$-...
7
votes
1
answer
2k
views
Crepant resolutions of cDV singularities?
Compound Du Val 3-fold singularities form a good class of singularities in 3-fold singularity theory. I would like to know which singularities admit crepant resolutions. If I remember correctly, $cA_{...
- 71
10
votes
3
answers
2k
views
Pullback along Frobenius morphism
Let $X$ be a scheme over a finite field $\mathbb{F}_q$ and let $F : X \to X$ be the absolute Frobenius morphism. If $\mathcal{L}$ is an invertible $\mathcal{O}_X$-module, then there is a natural ...
- 63.1k
3
votes
1
answer
484
views
Harish-Chandra isomorphism for characteristic $p$
I am trying to understand the proof of Theorem 1 from this paper V. Kac and B. Weisfeiler (Indag. Math. 1976, DOI link).
Theorem 1. Let either $p\neq 2$ or $\varrho\in X(\mathscr{T})$. Then $\gamma(...
- 385
4
votes
0
answers
119
views
Are there any explicit (prime-to-l) alterations for interesting varieties (or schemes)?
I have read that it is easier to find regular alterations of varieties than their resolutions of singularities (moreover, I believed in this sentence when I read it). My question is: do there exist ...
- 16.9k
1
vote
1
answer
247
views
Infinitesimal deformation of strict transform
Let $X$ be an affine, complex surface with isolated singularities and $\pi:\widetilde{X} \to X$ be a resolution of singularities (not necessarily minimal) i.e., $\widetilde{X}$ is non-singular and $\...
- 2,126
6
votes
1
answer
193
views
Restricted Lie algebras with no nonzero proper restricted subalgebras
Let $L\neq 0$ be a restricted Lie algebra over a field $F$ of characteristic $p>0$. If $F$ is algebraically closed, then it is known that $L$ has no nontrivial restricted subalgebras if and only ...
- 630
8
votes
2
answers
1k
views
Lefschetz on étale fundamental group for quasi-projective varieties
If $X$ is a smooth projective variety of dimension at least $3$ over $\mathbb{C}$, Lefschetz's Hyperplane theorem says that for every hyperplane section $H$
$$\pi^1(H)\to\pi^1(X)$$
is an isomorphism, ...
- 483
6
votes
1
answer
640
views
Resolution of Gorenstein rational singularities on a surface
I am reading Artin's notes "Lipman's Proof of Resolution of Singularities for Surfaces" from the book "Arithmetic Geometry". I am very confused by the proof of Lemma $6.5.$ (I am formulating it below ...
- 2,923
9
votes
1
answer
430
views
Existence of certain endomorphism of supersingular elliptic curve
Let $E$ be a supersingular elliptic curve over an algebraically closed field $K$ of characteristic $p$. Let $R = \operatorname{End}(E)$ be its ring of endomorphisms. Then, it is known $R \otimes_{\...
7
votes
1
answer
295
views
Do arithmetic schemes have non-singular alterations?
Let $X$ be an integral normal flat finite type scheme over $\mathbb{Z}$.
Does there exist a proper surjective generically finite morphism of schemes $Y\to X$ with $Y$ an integral regular ...
- 233
1
vote
1
answer
160
views
Generators of a graded algebra defining bundle over elliptic curve
I have a question about a statement from P. Wagreich's paper "Elliptic Singularities of Surfaces" (page 425):
We consider an elliptic curve $X$ and a line bundle (=invertible sheaf) $L$ on $X$.
Then,...
- 6,016
1
vote
1
answer
383
views
Construction of log canonical singularity
I know there's classification about normal log canonical surface singularity in the sense of configuration of exceptional curves.
There is one type of log canonical singularity(not klt) whose ...
- 623
5
votes
2
answers
426
views
Affinization of $T^*\mathbb{C}P^n$
Is there an elementary description of the affinization of the algebraic cotangent bundle of $\mathbb CP^n$? I know that it can be described as some sort coadjoint orbits, but I am interested in a ...
2
votes
2
answers
523
views
Crepant resolution of $Y=k[x,y,z]/(xz-y^3)$
Consider the action of $\mathbb{Z}_3\subset SL_2(k)$ on $\mathbb{A}^2$, we have the quotient $Y$ as in the title. According to the classification of Du Val singularity, we know that the crepant ...
- 815
2
votes
0
answers
85
views
Is toroidalization local?
Let $f:X \to Y$ be a surjective morphism of smooth projective varieties, $D$ be a simple normal crossings divisor on $X$ and $U_Y \subset Y$ be an open subset over which $(X,D)$ is log smooth (in the ...
- 393
3
votes
0
answers
143
views
What kind of equivalences exist between categories of characteristic $0$ and characteristic $p$?
The tilting equivalence for perfectoid algebras gives an equivalence of categories $$K\text{-perf} \cong K^\flat\text{-perf}$$
where the left-hand-side are algebras in characteristic zero and the ...
- 4,164
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
2
answers
917
views
Do fixed point sets in equivariant crepant resolutions have the same cohomology? How about for the specific case of Nakajima quiver varieties?
A crepant resolution $f:Y\to X$ is a resolution of singularities with $f^*(K_X)=K_Y$. Crepant resolutions do not always exist, and when they exist they may not be unique. However, different crepant ...
- 2,372
16
votes
2
answers
1k
views
Is the tangent space functor from PD formal groups to Lie algebras an equivalence?
The previous version of this question was rather badly broken, and I hope this version makes some sense.
There have been a few questions on MathOverflow about how much representation-theoretic ...
- 45.7k
4
votes
2
answers
340
views
On families of supersingular abelian surfaces over the projective line
Let $k=\mathbb{F}_q$. I recently learned that there are non-isotrivial families $f:X\to \mathbb{P}^1_k$ of supersingular abelian surfaces. In particular, the Kodaira-Spencer map of this family is non-...
- 43
3
votes
1
answer
190
views
Igusa zeta functions of univariate polynomials: $\mathbb{Z}_p$ or $\mathbb{Q}_p$ in this statement
Let $f\in\mathbb{Z}_p[X]$ and let $Z_{f,p}(T)\in\mathbb{Z}_{(p)}(T)$ be the $p$-adic Igusa zeta polynomial (i.e. $Z_{f,p}(p^{-s})$ is the $p$-adic Igusa zeta function in the complex variable $s$, with ...
18
votes
3
answers
3k
views
Lifting varieties to characteristic zero.
If you want to compute crystalline cohomology of a smooth proper variety $X$ over a perfect field $k$ of characteristic $p$, the first thing you might want to try is to lift $X$ to the Witt ring $W_k$ ...
- 4,015
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
5
votes
0
answers
217
views
Exact differential forms in characteristic $p>0$
Let $k$ be an algebraically closed field of characteristic $p>0$. Suppose $1< e_i <p$ for $i=1,2, \ldots, n$ are integers ($n \ge 2$). What are the conditions on the $e_i$'s so that the ...
- 245
2
votes
0
answers
163
views
Terminal and log canonical singularities
Let $D$ be a divisor with at most terminal singularities in a smooth projective variety $X$. Is the pair $(X,D)$ log canonical?
user61586
7
votes
3
answers
752
views
Does every linear group admit a subgroup of dimension 1?
Suppose that $G$ is a linear group of positive dimension, defined over some field $k$. Is that true, that $G$ admits a (closed) one-dimensional subgroup?
I'm pretty much sure this is true in ...
- 267
2
votes
0
answers
177
views
vanishing of higher algebraic de Rham cohomology and sheaves of differentials for singular curves
I'm looking for some results or references about de Rham cohomology of curves in less-than-optimal cases. The two vanishing results I care about are:
$H^{k}_{\text{dR}}(X) = 0$ for $k>2$, since $H^...
- 1,142
31
votes
4
answers
5k
views
The Frobenius morphism
I found the following list on the "Frobenius Page" by David Ben-Zvi, described by the author as "an outdated collection of intuitive ways to think about raising to the p-th power".
Generates a ...
22
votes
3
answers
2k
views
One dimensional (phi,Gamma)-modules in char p
I would like to better understand the simplest case of the correspondence between Galois representations and (phi,Gamma)-modules. Namely, consider 1-dimensional Galois representations of $G_{Q_p}$ ...
- 1,680
3
votes
0
answers
452
views
Singularities of rational quartic surfaces
Let $X\subset \mathbb{P}^3$ be an irreducible quartic surface, defined over an algebraically closed field $k$. Suppose that $X$ is rational (i.e. birational to $\mathbb{P}^2$). Is is true that $X$ has ...
- 7,722
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
1
vote
0
answers
125
views
Resolution of rational surfaces
Let $S$ be a rational singular complete algebraic surface over $\mathbb{C}$. Let $\phi:\tilde{S}\to S$ be a resolution of singularities with minimal possible Picard rank (i.e. minimal $\mathrm{dim}(...
- 4,189
6
votes
1
answer
1k
views
Normalization of complete intersection
Let $A$ be an integral complete local ring over a field which is complete intersection.
Let $B$ be a normalization of $A$.
Q. Is $B$ Gorenstein?
I guess that even the normalization of Gorenstein ...
- 781
19
votes
3
answers
2k
views
Elkies' supersingularity theorem in higher dimension
The following is a theorem of Elkies:
Let $X$ be an elliptic curve over $\mathbb{Q}$. Then there are infinitely many primes $p$ such that the action of Frobenius on $H^1(\mathcal{O}, X)$ is zero.
...
- 156k
3
votes
1
answer
351
views
Comparisons of log canonical thresholds
Premise
Let $K$ be a field of characteristic zero and $f\in K[X_1,\dots,X_m]$. By Hironaka's theorem, there exists a log resolution (over $K$) of the ideal $(f)$. Let $\{(N_i,\nu_i)\}_i$ be the ...
11
votes
1
answer
928
views
Non-algebraic K3 surfaces in characteristic $p$
I have a very naive question.
Recall that over the field of complex numbers, there exist non-algebraic K3 surfaces. Namely, smooth non-projective simply connected compact complex surfaces with ...
- 21.4k
1
vote
2
answers
508
views
Base change of a finite morphism
Let $X$, $Y$, $Z$ be integral schemes of finite type over a field $K$ (i.e., locally affine opens are finitely-generated algebras over $K$). Suppose we have the following condition$\colon$
$f \colon ...
- 563
3
votes
2
answers
369
views
Quotient of affine space by finite subgroup of SL(V) is Gorenstein
I am looking for a proof of the following fact:
If $G$ is a finite subgroup of $SL_n(\mathbb{C})$ acting on $\mathbb{A}_{\mathbb{C}}^n$, then the resulting quotient scheme is Gorenstein.
Thanks.
- 815
8
votes
1
answer
1k
views
Resolution of an isolated cyclic quotient singularity
I am looking for a reference to the following fact which seems to be true and which is probably well-known (at least to experts in resolution of singularities):
Consider an isolated cyclic quotient ...
- 165
4
votes
1
answer
330
views
example of quintics with 5 ordinary triple point
I know we can bound the triple point on quintics in cp^3 by 5. But how to write down quintics with 5 ordinary triple point (here are simple elliptic singularity)explicitly?
- 623
15
votes
1
answer
1k
views
Number of curves over a finite field
Let $K$ be a finite field. Is there a formula for the number of isomorphism classes of genus $g$ smooth curves over $K$?
In other words does there exists a formula for the number of rational points ...
- 8,998
5
votes
0
answers
568
views
Eisenbud-Goto conjecture in Positive Characteristic
Eisenbud-Goto conjecture predicted that the Castelnuovo-Mumford regularity
${\rm reg}(X)$ of a non-degenerate projective variety $X\subset \mathbb{P}^N$
is bounded by the $\deg(X)-{\rm codim}(X,\...
- 1,595
15
votes
1
answer
1k
views
Resolution of singularities in étale cohomology
The lack of a suitable resolution of singularities comes up often in work on étale cohomology from the 1960s and 70s, And I think even the latest version of Milne's lecture notes says "It is likely ...
- 11.2k
7
votes
2
answers
827
views
$p$-torsion of an abelian variety of $p$-rank $0$
Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $A$ be an abelian variety over $k$ such that $A[p](k) = 0$, i.e., such that $A$ has $p$-rank $0$. If I am not mistaken, ...
- 2,663
5
votes
1
answer
208
views
A generalization of Witt's theorem for quaternion algebra isomorphism
Let $Q$ be a quaternion $k$-algebra (namely, a dimension 4 $k$-central simple algebra).
Then it is possible to (canonically) attach a smooth projective conic $C_Q\subseteq \mathbf{P}_k^2$ to $Q$: if ...
- 375
10
votes
2
answers
761
views
Applications of alterations
In 1995, de Jong proved the existence of regular alterations in arbitrary characteristic. I would like to have a little survey of important applications of this theorem, e.g. things you could do if ...
- 1,072
14
votes
0
answers
1k
views
A slick proof (?) of Zariski-Nagata purity in characteristic $p$
I am trying to understand the MathSciNet review written by Mark Kisin of the paper "Almost etale extensions" of Faltings. There Kisin illustrates Faltings' approach to the almost purity theorem with ...
- 2,663
14
votes
3
answers
2k
views
Representations in characteristic p
Let G be a finite group and let F be an algebraically closed field. If the characteristic of F is 0, then the number of irreducible F-representations of G is given by the number of conjugacy classes ...
- 2,799