All Questions
624 questions
2
votes
1
answer
180
views
Locally toric resolutions of compactifications
Suppose $U$ is a smooth, open $n$-dimensional variety over $\mathbb{C}.$ Say $X, X'$ are two proper normal-crossings compactifications of $U$. Call a map $m: X'\to X$ a modification if it is an ...
- 4,111
2
votes
0
answers
121
views
Global invariant cycles in positive characteristic
Let $k$ be an algebraically closed field of positive characteristic $p$ and fix a prime $\ell\neq p$. Let $X$ be a smooth connected $k$-variety and $f:Y\rightarrow X$ a smooth projective morphism. ...
- 728
3
votes
0
answers
307
views
Isotrivial factors of Jacobian
Let $k$ be an algebraically closed field of positive characteristic that it is not the algebraic closure of a finite field. Fix a smooth proper $k$-curve $C$ and write $J_C$ for its Jacobian abelian ...
- 728
12
votes
2
answers
2k
views
Hitchin fibration and Springer resolution
Let C be a curve and let us assume $G=GL_N$ and $\mathfrak{g}=\mathfrak{gl}_N$ for simplicity. The moduli space $\mathcal{M}_H(C,G)$ of $G$-Higgs bundles admits the Hitchin fibration $\pi: \mathcal{M}...
- 1,073
3
votes
0
answers
113
views
Cohomologies of $[V/GL_n]$ in characteristic $p$ for a representation $V$ of $GL_n$
Let $V$ be a representation of $G=GL_n$ (or more generally any reductive group $G$) over an algebraically closed field $\mathbb k$ of characteristic $p$. Let $[V/G]$ be the corresponding quotient ...
- 790
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 ...
- 4,111
1
vote
0
answers
72
views
Minimal non-klt center of asymptotic linear system
Let $(X,\Delta)$ be a klt pair and $D $ a $Q $-Cartier divisor on $X $ such that the ring of sections of $D $ is finitely generated. Let $c$ be the log canonical threshold of the asymptotic linear ...
- 11
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
8
votes
0
answers
471
views
Sheaf whose singular support is not Lagrangian
For constructible sheaves $\mathcal F$ on real analytic manifolds $X$, there is a notion of the singular support $SS(\mathcal F)$ which is a radially invariant singular Lagrangian subset of the ...
- 18.7k
10
votes
0
answers
539
views
Why is resolution of singularities useful/important?
Why is it important/useful to resolve singularities generally, and in particular in algebraic geometry? I understand that one gets a nicer geometry, but how does it help one understand the geometry of ...
- 281
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?
- 79.9k
9
votes
1
answer
1k
views
Do the cohomology groups of the structure sheaf of a smooth resolution depend on the resolution?
Let $X$ be an affine variety. Let $Y$ be smooth and let the map $f\colon Y\rightarrow X$ be proper birational. We will call $Y$ a smooth resolution of $X$.
Do the cohomology groups $H^i(Y,\mathcal{O}...
- 4,111
27
votes
7
answers
6k
views
Etale covers of the affine line
In characteristic p there are nontrivial etale covers of the affine line, such as those obtained by adjoining solutions to x^2 + x + f(t) = 0 for f(t) in k[t]. Using an etale cohomology computation ...
- 8,866
24
votes
5
answers
6k
views
Wild Ramification
The question is, loosely put, what is known about wild ramification?
Is there a semi-well-established theory of wild ramification that can be furthered in various specific situations? Or maybe there ...
- 153
5
votes
1
answer
172
views
Resolving $\mathbb Z_n$ action on $\mathbb C^2$
Consider a diagonal action of $\mathbb Z_n$ on $\mathbb C^2$ generated by $(z_1,z_2)\to (\mu^pz_1,\mu^qz_2)$, with $\mu^n=1$.
Question. Is it always possible to find a smooth blow up $X\to \mathbb ...
- 6,253
3
votes
0
answers
269
views
surjectivity of double dual map for weil divisors on normal varieties
If $X$ is a normal complex variety, and $D$ is an effective $\mathbb{Q}$-Cartier Weil divisor, then there is a natural map $\mathcal{O}(D) \otimes \mathcal{O}(-D) \rightarrow \mathcal{O}_X$. My ...
- 83
3
votes
0
answers
352
views
smoothing of isolated surface singularity
I want to know when an isolated surface singularity can be smoothed, especially for log canonical isolated surface singularity. Is there any good reference. Thanks in advance.
- 623
1
vote
0
answers
348
views
rigid analytic geometry positive characteristic
I am a beginning graduate student. I have the following basic question I am very confused about:
Suppose $C$ is a smooth geometrically irreducible curve over a finite field $\mathbb{F}_q$, $q=p^m$, $...
- 147
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 ...
- 7,800
3
votes
1
answer
410
views
Derivations of central extensions of simple Lie algebras
Let $L$ be a finite-dimensional Lie algebra over a field of characteristic zero. It is not difficult to see (and also follows from Theorem 4.4 of [G. Hochschild: Semi-simple algebras and generalized ...
- 5,878
3
votes
0
answers
448
views
simple elliptic surface singularity
Suppose X is a one dimension torus and L is a line bundle over X, I think one class of log canonical surface singularity comes from the contraction of one elliptic curve from the total space L. My ...
- 623
17
votes
2
answers
1k
views
Higher level analogs of Nicolas-Serre theory
NICOLAS-SERRE THEORY
Let $F \in Z/2[[x]]$ be $x+x^9+x^{25}+...$, the exponents being the odd squares, and $V$ be the space spanned by the $F^k$ with $k$ odd. Nicolas and Serre define formal Hecke ...
- 5,422
3
votes
1
answer
270
views
Restriction of separable map
If $f: X\to Y$ is a separable map between varieties that is a bijection on closed points, is it true that $f$ remains separable when restricted to an integral subscheme $Z\subset X$?
If we drop the ...
- 4,111
2
votes
0
answers
214
views
log canonical surface singularity
For a log cannonical surface singularity, I guess there is classification about the configuration of the exceptional divisor of its minimal resolution. I wonder if there is some specific example or ...
- 623
3
votes
0
answers
82
views
Singularities of fibrations 2
This question is related to my previous question:
Singularities of fibrations
Assume that $X$ is a complete intersection irreducible $3$-fold in a product of projective spaces. So that $X$ is ...
- 8,998
4
votes
1
answer
327
views
Singularities of fibrations
Let $f:X\rightarrow \mathbb{P}^2$ be a fibration, here $X$ is a projective variety of dimension three.
Assume that there exixts a smooth curve $C\subset\mathbb{P}^2$ such that for any $p\in\mathbb{P}...
- 42.9k
2
votes
1
answer
167
views
Singularities of $3$-folds
Let $X,Y,Z$ be projective $3$-folds. Assume that $Y$ is smooth and $Z$ is smooth and Fano. Moreover, assume that there is a generically finite morphism $f:Y\rightarrow Z$ admitting a factorization $f=...
- 20.5k
5
votes
0
answers
319
views
Resolving analytic normal crossings singularities
Let $X$ be a non-singular (complex) variety and $Y \subset X$ be a (reduced) irreducible subvariety with only normal crossings singularity (locally, in the analytic topology, the singularity is ...
- 1,981
3
votes
0
answers
157
views
Resolving complete intersections of quadrics with singularities
Suppose that $X$ is a complete intersection of quadrics in $\mathbb P^n_{\mathbb C}$. Is there some straightforward procedure to resolve the singularities of $X$?
For example, can one stratify ...
- 14.3k
3
votes
1
answer
343
views
Two spaces attached to mod 2 level 9 modular forms--a conjectural Hecke isomorphism
MOTIVATION
Nicolas and Serre have analyzed the structure of the space of mod $2$ modular forms of level $1$, viewed as a "Hecke-module". They show that for each $p>2$, the operator $T_p$ acting on ...
- 5,422
2
votes
0
answers
196
views
Total space of canonical bundle on toric del pezzo surface
Let $X$ be a toric del pezzo surface with a full cyclic strong exceptional collection of line bundles, say $\mathbb{E}:=(E_1,\ldots,E_n)$, consider the total space of anti-canonical bundle $\omega_X^*$...
2
votes
0
answers
286
views
Does the sheaf of locally exact differential forms splitting in positive characteristic
Let k be an algebraically closed field of characteristic $p>2$, $X$ a smooth projective curve of genus $g>1$ over $k$, and $F_X:X\rightarrow X$ be the absolute Frobenius morphism. Let $B^1_X$ be ...
- 85
6
votes
0
answers
343
views
Are all stabilizer groups of the co-adjoint action smooth?
Let $k$ be a (non-archimedean) local field of positive characteristic $p$ and $\mathfrak{n}$ be any finite-dimensional nilpotent Lie algebra over $k$ with nilpotence length $l<p$. It is well-known ...
CommunityBot
- 1
6
votes
0
answers
467
views
Torsionfree crystalline cohomology implies torsionfree etale cohomology?
Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic $p>0$. Let $W=W(k)$ be the ring of Witt vectors of $k$.
Assume that the crystalline cohomology $H^2_{...
- 91
1
vote
1
answer
241
views
locally closed orbits in metric Hausdorff topology
I learned the following fact from Bruhat and Tits's paper "Homomorphismes “abstraits” de groupes algebriques simples" Section 3.18 that
Let $k$ be a local field. Suppose that a $k$-group $H$ acts $k$...
- 1,702
1
vote
0
answers
74
views
Simple question about surface singularities
Given $\epsilon \in (0,1)$, is it possible to find two finite familes $\mathcal{F}$ and $\mathcal{P}$ of weighted graphs, such that the weighted graph of the minimum resolution of any $\epsilon$-klt ...
- 1,595
7
votes
0
answers
177
views
How to compactify $\mathbb{Z}_p[x, y, z]/(xyz - p)$?
The affine scheme $U := \mathrm{Spec}(\mathbb{Z}_p[x_0, \ldots, x_d]/(x_0\cdots x_d - p))$ is regular and is a basic example of a semistable scheme over $\mathbb{Z}_p$. How does one build a proper ...
- 2,663
1
vote
0
answers
187
views
Unitary dual of the Heisenberg group over non-archimedean local fields of characteristic two
What is the unitary dual of the Heisenberg group over non-archimedean local fields k of characteristic two? This is well-known for the real Heisenberg group and in fact, when local fields have ...
- 1,702
3
votes
1
answer
684
views
Canonical sheaf of affine variety
Let $A=\mathbb{C}[u,x,y,w]/(uy-x^2,xw-y^2,uw-xy)$, $X=Spec A$. $A$ is a Veronese subring and from the answer of Is there a simple method to test a local ring to be Cohen Macaulay?, we can see that $X$ ...
CommunityBot
- 1
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 ...
CommunityBot
- 1
12
votes
1
answer
750
views
Vanishing theorems in positive characteristic
In the paper
Deligne, Pierre; Illusie, Luc (1987), "Relèvements modulo $p^{2}$ et décomposition du complexe de De Rham", Inventiones Mathematicae 89 (2): 247–270, doi:10.1007/BF01389078
I found the ...
- 66.3k
7
votes
0
answers
374
views
Arbitrarily non-degenerate Hodge to de Rham spectral sequence
It is true that for any $n$ there exists a compact complex manifold which Frolicher spectral sequence does not degenerate at the $n$-th page(https://arxiv.org/pdf/0709.0481.pdf).
Does the analogous ...
- 7,377
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{...
CommunityBot
- 1
5
votes
2
answers
1k
views
Weil Conjectures for Grassmannians
To establish the Weil conjectures for $n$-dimensional projective space over a finite field is elementary. Does there exist a simple direct proof of the conjectures for finite field Grassmannians?
- 2,218
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 ...
- 4,111
4
votes
1
answer
269
views
Functors between equivariant derived category and derived category of the quotient
Suppose $M$ is a quasi-projective variety, $G$ is a finite group acting on $M$. Let $X$ be the quotient $M/G$ (we assume $X$ to be singular) and $\pi: M\to X$ be the natural projection.
We have $(\pi)...
- 815
2
votes
0
answers
230
views
(Strict) canonical singularity with no crepant resolution
We know if a normal singular $\mathbb{Q}$-factorial variety $X$ has a crepant resolution, then it is canonical but not terminal. What can we say about the converse? To be more precise:
Is there a ...
- 815
1
vote
0
answers
149
views
Smoothability of stable curves in mixed characteristic
Let $R$ be a complete DVR with residue field $k$ algebraically closed of characteristic $p$ and fraction field $K$ of characteristic zero. Let $C$ be a stable curve (in the sense of Mumford-Deligne) ...
- 2,323
-1
votes
1
answer
99
views
Finitely many subvarieties as divisor
Let $X$ be a smooth projective variety over an algebraically closed field of characteristic $0$ and of dimension $n\geq 10$. Let $(C_i)_{1\leq i\leq N}$ (resp. $(S_i)_{1\leq i\leq N}$) be smooth ...
2
votes
0
answers
345
views
Examples of semi-stable models of curves
Let $R$ be a discrete valuation ring with fraction field $K$ of characteristic zero and residue field $k$ of characteristic $p>0$. Assume $k$ is algebraically closed. I want to produce examples of ...
- 2,323