All Questions
512 questions
1
vote
1
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280
views
Norms of elements in Artin-Schreier extensions
The following is claimed in the proof of Theorem 7.5 of Auslander, Goldman, "The Brauer group of a commutative ring":
Let $k$ be a nonperfect field of positive characteristic $p$, let $K := k(x)$ ...
- 428
17
votes
2
answers
1k
views
A short proof for simple connectedness of the projective line
The Riemann-Hurwitz formula implies that the projective line $\mathbb{P}^1_K$ over any algebraically closed field $K$ is simply connected (i.e., $\pi_1^{et}(\mathbb{P}^1_K) = 1$; equivalently, if $\...
- 3,362
2
votes
0
answers
304
views
Surjectivity of map of Picard schemes implies abelian
Note: This question was asked on MSE first, but got zero reactions. So I deleted it there, and am now posting it here.
I am looking for a reference or explanation of the fact that is used in Mumford'...
- 203
2
votes
0
answers
121
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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
7
votes
0
answers
571
views
Pseudo-effective divisor which is not nef in any birational model
Let $X$ be a smooth complex projective algebraic variety and let $D$ be a $\mathbb{Q}$-Cartier pseudo-effective divisor on $X$. Lets say that $D$ is birationally nef
if there exists a birational ...
- 1,595
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
11
votes
1
answer
334
views
An identity in Lie algebras over fields of positive characteristic
Let $L$ be a Lie algebra over a field of characteristic $p>0$ and $D$ a derivation of $L$. For every $x\in L$ denote by $\mathrm{ad} x$ the adjoint map $\mathrm{ad}x: L \rightarrow L, a\mapsto [x,...
- 630
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
7
votes
0
answers
404
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Finding basis of cohomology of a symplectic manifold by using Symplectic Minimal Model Program
My question is about Floer theory via symplectic surgery of Minimal Model program for finding basis of cohomology.
Motivation: Perelman for solving Thurston's Geometrization Conjecture used some sort ...
user21574
3
votes
0
answers
78
views
Finiteness of models around a non-pseudo-effective ray
Let $(X,\Delta) $ a klt pair and $\rho $ a numerical class of divisor that is not contained in the pseudo-effective cone. Let $(X,\Delta_i)$ be a sequence of klt pairs such that $K_X+\Delta_i$ is not ...
- 1,595
1
vote
1
answer
927
views
relative tangent sheaf
Let $f:X\rightarrow Y$ be a surjective birational morphism of varieties. Suppose the center of the birational morphism is $Z$ and $f:f^{-1}(Z)\rightarrow Z$ is a $\mathbb{P}^n$-bundle. Consider the ...
user100841
5
votes
1
answer
1k
views
Bertini's type theorems over imperfect fields
Let $X$ be a projective variety over an imperfect (hence infinite and char(k)=p>0) field $k$. If the local rings of $X$ are all regular, then can we say that a general hyperplane section $H$ is also ...
- 313
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
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
4
votes
0
answers
180
views
Deminormal and Gorenstein
Let X be an irreducible deminormal variety such that the normalisation is Gorenstein. Does it follow that X is also Gorenstein?
for deminormal definition, see https://arxiv.org/pdf/1506.02002.pdf
- 151
0
votes
1
answer
242
views
dualizing sheaf of deminormal variety
Let X be a deminormal variety (over Char 0). Then is it true that the dualizing sheaf is divisorial?
Please provide a reference..
- 151
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
9
votes
1
answer
1k
views
deformation theory in positive characteristic
The idea "Formal deformation theory in characteristic zero is controlled by a differential graded Lie algebra (dgla)" goes back to Goldman-Millson, Deligne, Drinfeld among others; see Lurie's ICM talk....
- 528
4
votes
0
answers
221
views
A question about the dimension of a relatively ample divisor
Suppose $f: Y \to Z$ is a projective morphism of smooth varieties with connected fibers. If an effective divisor $H$ on $Y$ is relatively ample over $Z$, and $\dim Y >\dim Z$, is $h^0(Y, mH)>1$ ...
- 3,472
4
votes
0
answers
182
views
Kuranishi family and smoothing of Calabi-Yau n-fold
Consider $X$ be a Calabi-Yau n-fold with at
most one ordinary double point singularity. Suppose $X$ is smoothable. Then it is known that the Kuranishi family of $X$ is a smoothing of $X$.
Now, ...
- 81
4
votes
0
answers
218
views
Example of a non-algebraic singularity II
In an answer of this MO question, Frank Loray constructed an example of analytic singularity which is not algebraic. On the other hand, as I learned from one of Joël's comments in that question, ...
- 1,829
3
votes
0
answers
290
views
Nefness property for symplectic equivalency of Moishezon manifolds
Definition:Two symplectic manifolds $(X,\omega_X)$ and $(Y,\omega_Y)$ are defined to be symplectically equivalent if there exists a diffeomorphism $\phi:X\to Y$ such that $\phi^∗ω_Y$ is in the same ...
user21574
2
votes
0
answers
442
views
Moishezon projectivity criterion for Moishezon spaces with canonical singularites
A Moishezon manifold is projective if and only if it is Kähler. This is no longer true for a singular Moishezon space. Moishezon proved a projectivity criterion for Moishezon spaces with isolated ...
user21574
6
votes
1
answer
519
views
Picard number of a general fiber of a fiber contraction
Suppose in the last step of a MMP, we obtain a Mori fiber space $f: X \to Z$, and let $F$ be a general fiber of $f$, then is the Picard number $\rho(F)$ of $F$ equal to $1$? Notice that the relative ...
- 3,472
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
6
votes
0
answers
564
views
Bogomolov–Miyaoka–Yau inequality for minimal varieties with intermediate Kodaira dimension $0<\kappa (X)<\dim X$
From the differential geometric proof of Yau and the algebraic proof of Miyaoka for minimal varieties of general type $\kappa (X)=\dim X$, we know that $$(-1)^nc_1^n(X)\leq (-1)^n\frac{2(n+1)}{n} c_1^...
user21574
2
votes
1
answer
715
views
Castelnuovo and Artin contractibility criteria for families
In the course of a proof, I need some version of Castelnuovo and Artin's contractibility criteria for a family of surfaces. Say I have a family (flat is probably needed, in order to compare ...
- 625
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 ...
- 1,537
8
votes
1
answer
318
views
rational effective implies effective?
Let $X$ be a weak del pezzo surface, I Wonder whether the following statment is true:
Let $L$ be a line bundle on $X$, then $h^0(L)=0$ implies $h^0(nL)=0$ for all $n\geq 1$.
- 1,982
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
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 ...
- 1,702
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
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
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
6
votes
1
answer
617
views
Advantage of discrepancy
In the definition of Minimal model of projective variety, some authors use of discrepancy, and some others omit this condition. I am wondering to know the advantage of discrepancy In the definition of ...
- 63
4
votes
0
answers
628
views
A theorem about log Calabi-Yau pairs
Let $X$ be a normal variety with $\mathbb Q$-Cartier divisor $D$, such that $K_X+D$ is $\mathbb Q$-Cartier. Let $(X,D)$ is log Calabi-Yau pair, i.e, $K_X+D\sim_\mathbb Q0$. (For example take $ X$ be a ...
user21574
3
votes
1
answer
711
views
Adjunction formula on pair
Assume $X$ be a normal projective variety with $\mathbb Q$-Cartier divisor $D$, then can we extend adjunction formula on pair $(X,D)$?
- 33
7
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0
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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
0
votes
1
answer
119
views
holomorphic fiber space when fibers are of general type
Let $\pi:X\to S$ be a holomorphic fibre space, then from birational geometry, the canonical divisor of general fibers are ample or trivial. When fibers are of general type?
- 75
3
votes
0
answers
198
views
$L^2$ extension theorem
Is there an Ohsawa-Takagushi $L^2$-Extension theorem for Kahler manifolds? For projective varieties Siu-Paun proved:
Let $\pi \colon X \to \mathbb D$ be a projective family of $n$-folds and $X_0$ be ...
- 99
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{...
user19475
2
votes
0
answers
237
views
Top self-intersection of the canonical divisor of a terminalization
Let $(X,\Delta)$ be a $n$-dimensional log canonica model,
and let $Y\rightarrow (X,\Delta)$ be a log terminalization
(meaning that $Y$ is the model obtained when we take a log resolution $\pi \colon ...
- 1,595
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
4
votes
0
answers
172
views
A question about potentially birational divisor
I am reading the paper "On the birational automorphisms of varieties of general type", and I have a question about the property of potentially birational divisor.
Definition (potentially birational ...
- 3,472
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
4
votes
1
answer
272
views
Finiteness of cohomology with finite coefficients
Let $G$ be a finite abelian group and let $S$ be a variety over $\mathbb{F}_p$. It is natural (I think) to expect that the cohomology group $H^i(S,G)$ is finite. But with respect to which cohomology?
...
- 41
8
votes
2
answers
1k
views
References for the minimal model program
What are some references for a beginning graduate student in algebraic geometry to learn about the minimal model program? I'm not thinking about entering this field, but rather I just want to know ...
- 1,537
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
2
votes
0
answers
148
views
Purely inseparable $k$-rational dominant maps between an absolutely irreducible $k$-surface and $\mathbb{P}^2$
Let $X$ be an absolutely irreducible surface over an algebraically closed or finite field $k$ of characteristic $p$. Is it true that the following conditions are equivalent?
There is a purely ...
- 1,833