All Questions
512 questions
9
votes
2
answers
2k
views
The minimal model program and symplectic resolutions
I've been reading some papers of Namikawa lately, and have on several occasions come across a claim I would really like someone to expand on.
On page 4 of Poisson deformations of affine symplectic ...
- 44.7k
14
votes
2
answers
1k
views
Can a reductive group act non-linearly on a vector group?
Let $k$ be a field; I'm going to discuss linear algebraic groups over $k$. The question I'll pose is only interesting when the characteristic is $p>0$.
1. Some motivation
A vector group is an ...
- 3,246
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
2
votes
0
answers
476
views
Does the numerical equivalence relation coincide with the homological one for 1-cycles (in positive characteristic)?
Is the Grothendieck's Standard Conjecture D (stating that the numerical equivalence relation for algebraic cycles with rational coefficients coincides with the homological one) known to be true for ...
- 16.9k
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
1
vote
0
answers
50
views
How can I describe in explicit geometric terms the (in general non-complete) linear system?
Let $\varphi\!: S \to S^\prime$ be a birational morphism of projective non-singular irreducible surfaces over a (algebraically closed) field $k$ and let $D \in \mathrm{Div}(S)$. Also, let $\big(\...
- 1,833
7
votes
1
answer
5k
views
Chevalley's Theorem on Constructible Sets
I'm having a hard time understanding the theorem in the title, more specifically the proof of the related fact that the image of a dominant morphism contains a dense open set of it's closure. (My ...
- 75
19
votes
1
answer
1k
views
Is there a connected $k$-group scheme $G$ such that $G_{red}$ is not a subgroup?
I've been trying a learn a little more about group schemes by working through a set of exercises on Brian Conrad's website. Exercise 8.3 of http://math.stanford.edu/~conrad/papers/gpschemehw1.pdf ...
- 3,625
1
vote
2
answers
550
views
Decompose a big divisor as nef big divisor and effective divisor
Let $W_n$ be a set of a log pair having the following property:
For any $(X, D) \in W_n$
(1)$X$ has dimensional $n$ with tirvial canonical divisor (i.e.$K_X = 0$). Moreover, $X$ is a $\mathbb{Q}$-...
- 3,472
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
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
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
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
4
votes
0
answers
235
views
Some examples where the plurigenera are nonconstant, when the fibres have worse singularities than canonical
Let start with a definition
Invariance of plurigenera: Choose $m$ large enough so that $mK_F$ has a non-zero global section for some fibre $F$. For any fibre $F$, we have $K_F = K_{X/D}~_{|F}$. So ...
user21574
10
votes
1
answer
570
views
Commutativity of the Chow ring in positive characteristic
I was looking in Ravi Vakil's notes on Intersection Theory, Class 20, where he introduces the bivariant intersection theory, in particular the Chow ring $A^\ast (X)$.
On p. 2, he writes the following ...
- 40.3k
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
3
votes
1
answer
500
views
Run MMP between varieties of isomorphic in codimension 1
Let $X, Y$ be two birational projective varieties which are isomorphic in codimension 1. Suppose $H$ is an ample divisor on $Y$, and $H'$ be its strict transform on $X$, suppose we can run MMP with ...
- 3,472
2
votes
2
answers
480
views
Lifting to char 0, references and questions
Suppose that I have a surface $S$, smooth proper over an algebraically closed (perfect?)field $k$ that lifts algebraically to some $S_W$ defined over a field of char 0. I am interested in properties ...
- 645
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
5
votes
0
answers
257
views
Symplectic leaves in positive characteristic
I am currently considering a family of filtered algebras over an algebraically closed field of positive characteristic with the property that the associated graded algebra is a finitely generated ...
- 599
7
votes
1
answer
609
views
Is there an Enriques–Kodaira-like classification of Fano threefolds?
I am mainly interested in varieties over an algebraic closed field $k$ (or $\mathbb{C}$). The classification of complex surface is established in the last century and known as Enriques–Kodaira ...
- 73
4
votes
0
answers
214
views
Some questions on Kontsevich's moduli space
Motivation: Work of Eisenbud, Harris, and Mumford shows that
$\mathcal M_g$ is of general type when $g≥24$. Moreover, by Logan's function $f(g)$ , $\overline {\mathcal M_{g,n}}$ is of general type for ...
user21574
7
votes
4
answers
736
views
Simply connected quasi-projective varieties in positive characteristic
I am looking for examples of non-projective (quasi-projective) varieties $X$ defined over a field of positive characteristic, which have trivial étale fundamental group.
It is well known that the ...
- 4,450
11
votes
1
answer
2k
views
Are automorphism groups of hypersurfaces reduced ?
In the following article : "H. Matsumura, P. Monsky, On the automorphisms of hypersurfaces, J. Math. Kyoto Univ. 3 (1964) 347-361", it is shown that in finite characteristic, automorphism groups of ...
- 6,543
2
votes
1
answer
915
views
Properties of extreme rays
Let $X$ be a projective variety over $\mathbb{C}$, then the effective curves module the numerical equivalence form a cone. For my understanding, if the extreme ray $[C]$ has the property that $[C]\...
- 3,472
15
votes
1
answer
4k
views
Frobenius Descent
Let $S$ be a scheme of positive characteristic $p$ and $X$ a smooth $S$-scheme. Let $F:X\rightarrow X^{(p)}$ denote the relative Frobenius. A result by Cartier (often called Cartier descent or ...
- 4,450
9
votes
1
answer
502
views
Is the number of minimal models finite
Let $X$ be a variety of general type.
Assume that $\dim X = 3$. In https://eudml.org/doc/164223 it is proven that $X$ has only finitely many minimal models (i.e., only $\mathbb Q$-factorial terminal ...
- 133
6
votes
1
answer
211
views
Recursions for some binary theta series in characteristic 3
Define $A(0), A(1), A(2) \dots$ in ${\bf Z}/3[[x]]$ as follows. For $n$ in $\bf N$ let $s=3^{2n+1}$. Then $A(n) = \sum a_kx^k$ where $a_k$ is the mod 3 reduction of the number of representations of $k$...
- 5,422
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
0
votes
0
answers
152
views
$C^\infty$-curvature of Kawamata's singular hermitian metric
Let $X,Y$ are two projective varieties and $f:X\to Y$ is an Iitaka fibration. Consider the following singular hermitian metric $$h(\sigma,\sigma)=\left(\int_{X_y}|\sigma|^{\frac{2}{m!}}\right)^{m!}$$ ...
user21574
6
votes
1
answer
693
views
Nagata's conjecture in positive characteristic
For a $\mathbb C P^2$ is known a result: if through the generic points $p_1,p_2,\dots p_n$ with multiplicities $m_1,m_2\dots, m_n$ correspondingly a degree $d$ irreducible reduced curve passes then $d^...
- 5,063
4
votes
2
answers
394
views
Colon property of Gorenstein rings
Let $(R, \mathfrak{m})$ be a Gorenstein local ring of characteistic $p>0$. Let $x_1,...,x_d$ be a system of parameters of $R$. Let $I$ be an $\mathfrak{m}$-primary ideal containg $(x_1,...,x_d)$. ...
- 2,773
3
votes
1
answer
391
views
Embedded resolution of curves on smooth varieties
As far as I understand, embedded resolution of singularities means the following: given a variety $X$ over an algebraically closed field, and a closed subvariety $Y$, there exists a birational map $f:...
- 16.2k
15
votes
0
answers
2k
views
Why was it so difficult to define the relative de Rham-Witt complex?
In Illusie's original article, the de Rham-Witt complex is defined for a smooth scheme over a perfect characteristic $p$ base $S$, without reference to $S$. Some 25 years later, Langer and Zink ...
- 16.2k
4
votes
1
answer
998
views
Dimension of irreducible representations in characteristic p
Hi, I know that over $\mathbb{C}$ the dimension of an irreducible representation of a finite group $G$ must divide the order of the group. I've read somewhere that if $p$ does not divide the order of ...
- 407
2
votes
1
answer
727
views
Why are the different definitions of minimal model equivalent?
I'm starting to learn the minimal model program. It seems there are two definitions for a variety $X$ with only terminal singularities to be minimal:
$K_X$ is nef.
Every birational morphism from $X$ ...
- 93
5
votes
1
answer
1k
views
Minimal Model Program for surfaces over algebraically closed fields of characteristic p
Let $k$ be an algebraically closed field of characteristic $p>0$.
I have been trying to find out unsuccessfully if there is a mmp for algebraic surfaces over $k$. I know minimal surfaces are ...
- 2,215
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
1
answer
674
views
Inseparable Galois Cohomology
First let me give a general form of my question, and then I'll give some motivation and a more specific version of it. Let $K/k$ be a Galois extension of fields with Galois group $G$, and let $X$ be ...
- 1,199
6
votes
2
answers
945
views
Notation/name for "Artin-Schreier roots"?
If x is an element of a field K and n is a positive integer, we have both a symbol and a name for a root of the polynomial t^n - x = 0: we denote it by x^{1/n} and call it an nth root of x.
Of course ...
- 65.4k
15
votes
2
answers
814
views
Can the failure of the multiplicativity of Euler factors at bad primes be corrected?
Warning: This one of those does-anyone-know-how-to-fix-this-vague-problem questions, and not an actual mathematics question at all.
If $X$ is a scheme of finite type over a finite field, then the ...
- 9,418
3
votes
1
answer
254
views
Finite generation of certain $\mathcal{O}_X$-algebra
It is proved in this paper by Kawamata (Theorem 6.1) that for a 3-dimensional normal algebraic variety $X$ which has at most canonical singularities, and a Weil divisor $D$ on it, the $\mathcal{O}_X$-...
- 3,472
17
votes
0
answers
1k
views
Katz--Mazur for abelian varieties
Over $\mathbb Z$, there is a smooth DM stack $A_g$ classifying abelian varieties.
Over $\mathbb Z[\frac 1N]$, there is finite etale cover $A_g(N)_{\mathbb Z[\frac 1N]}\to A_g\otimes\mathbb Z[\frac 1N]...
- 18.7k
4
votes
1
answer
398
views
Properties of divisors when moving from char 0 to char p.
Consider a smooth projective variety $X$ over $\mathbb{C}$ such that $X$ has models over $\mathbb{Z}[1/N]$ and $X_p=X_{\mathbb{Z}[1/N]}\times \text{Spec}(\mathbb{F}_p)$ is also a smooth projective ...
- 141
7
votes
0
answers
355
views
Are curves over imperfect fields defined over a smaller field?
Let $C$ be regular projective curve defined over a field $K$. Let $K/L$ be a totally inseparable finite extension. Does there exist a regular projective curve $C'$ over $L$ such that that the pullback ...
- 149k
6
votes
1
answer
804
views
Del pezzo surfaces in positive characteristic
For me a Del Pezzo surface $X$ over an algebraically closed field of characteristic $p$ is an algebraic surface where the anticanonical bundle $\omega^{-1}_X$ or $-K_X$ is ample. (I prefer the second ...
- 2,215
8
votes
0
answers
381
views
Degeneration of wildly ramified local monodromy representations - near or far from Deligne?
Suppose we have a surface $S$ mapping to a curve $C$ and a finite cover $Y/S$ that is ramified at a divisor $D$. For each point $x \in C$ we get a ramified cover of the curve $S_x$, and we can study ...
- 149k
10
votes
0
answers
573
views
Singularities arising from the Minimal Model Program (an algebraic point of view)
I will start the story by the end:
Is there some characterization of (some of) the singularities arising from the Minimal Model Program (canonical, terminal, log-...) in terms of commutative algebra ?...
- 987
7
votes
3
answers
3k
views
congruent to 1 mod p
This is a somewhat vague question: for a prime number p, we often see that various counts come out to be 1 modulo p. What are the possible reasons for this?
Here are some I've encountered:
For some ...
5
votes
3
answers
739
views
Smoothness of hyperplane sections
Suppose $X\subset \mathbb{P}^n$ is a smooth hypersurface defined over $\mathbb{Q}$. For a "generic" prime $p$, what can be said about the set of hyperplanes $H$ in $\mathbb{P}^n(\mathbb{F}_p)$ for ...
- 13.1k