All Questions
1,159 questions
4
votes
0
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814
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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
-1
votes
1
answer
422
views
on flat morphisms
Let $j:U\rightarrow X$ an open immersion between k-schemes of finite type and $f:X\rightarrow S$ a surjective k-morphism of finite type.
We suppose that $f\circ j:U\rightarrow S$ is faithfully flat, ...
- 3,472
4
votes
2
answers
2k
views
When is the determinant of the push-forward of an ample line bundle ample
Let $f:X\to S$ be a "nice" morphism of "nice" schemes. Let $L$ be an ample line bundle on $X$.
When is $\det f_\ast L$ also ample?
A "nice" morphism could be anything from "finite type separated" to ...
5
votes
2
answers
586
views
Quotient of a reductive group by a non-smooth subgroup
This is a continuation of my question Quotient of a reductive group by a non-smooth central finite subgroup.
Let $G$ be a smooth, connected, reductive $k$-group over a field $k$ of characteristic $p&...
- 14.2k
1
vote
0
answers
301
views
How to Construct a ''Nice'' Birational Model in Characteristic $p>0$?
Let $X={\rm Spec} A$ be a normal affine variety of dimension $n$ over an algebraically closed field $k$ of characteristic $p>0$. Also, assume that $S\subset X$ is a prime Weil-divisor on $X$.
Now,...
- 165
24
votes
2
answers
2k
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Have we ever proved any non-solvable case of reciprocity without the Langlands program ?
The reciprocity of the title is the following not completely well-posed problem:
Fix $P(X)$ a monic irreducible polynomial of degree $n$, with coefficients in $\mathbb Z$. "Describe"
(in some sense) ...
- 26.1k
10
votes
1
answer
841
views
Is there a direct proof that affine schemes are fppf quasi-compact?
Let $A$ be a (commutative) ring. A family $(B_i)_{i\in I}$ of $A$-algebras is said to be an fppf cover if it satisfies three properties: (1) each $B_i$ is flat as an $A$-module, (2) each $B_i$ is ...
- 9,418
3
votes
1
answer
642
views
Decomposition theorem for polarized abelian varieties in positive characteristic
In characteristic zero we have the following decomposition theorem for polarized abelian varieties: it gives an isomorphism between a PPAV and a product of PPAV's of lower dimension and is valid (as ...
- 614
8
votes
1
answer
741
views
What about schemes built up out of graded rings?
Toen-Vaquié construct a category of schemes relative to some complete cocomplete closed symmetric monoidal category $C$. Affine schemes correspond by definition 1:1 to commutative monoid objects in $C$...
- 63.1k
5
votes
0
answers
387
views
Are these empirical discoveries about the Serre Swinnerton-Dyer ring of prime level modular power series actual theorems?
In this question Joel Bellaiche constructed an algebra, M, of modular forms for
gamma_0 (N) in finite characteristic (which he called p, but I'll call ell) and asked to know its structure. Matt ...
- 5,422
2
votes
0
answers
175
views
Regular subscheme of a projective limit of schemes
Let $S\cong \varprojlim S_i$, where $S$ and all $S_i$ are separated regular excellent of finite Krull dimension. Let $Z$ be a closed regular subscheme of $S$. As Theorem 8.8.2 of EGA4 shows, $Z$ comes ...
- 16.9k
2
votes
1
answer
303
views
Lifting vector fields to its resolution in char $p$
In this paper 4.7, the authors showed that on a normal variety $X$, if there is a tangent vector field on its smooth locus, then it can be lifted as a logarithmic tangent vector field on a log ...
- 656
4
votes
1
answer
537
views
Which schemes can be presented as limits of smooth varieties?
I can prove a certain statement for any scheme that can be presented as the limit of an essentially affine (filtering) projective system of smooth varieties over a perfect field such the connecting ...
- 16.9k
2
votes
0
answers
143
views
Dualizing sheaf in mixed characteristic for regular schemes.
I've been looking many places, but everything I find seems to either talk about (a) varieties or (b) extremely general situations with dualizing complexes. As I am not in the situation of (a) (i.e. ...
- 419
2
votes
1
answer
307
views
On a Strongly F-regular Pair (X, \Delta)
Let $X$ be a normal projective variety over a field of characteristic $p>0$ and $(X, \Delta\geq 0)$ be a pair such that $K_X+\Delta$ is $\mathbb{Q}$-Cartier whose index is not divisible by $p$. ...
- 165
0
votes
0
answers
259
views
How would you call a subscheme of a smooth $S$-scheme?
In my preprint I propose to call $X/S$ quasi-smooth if $X$ can be embedded into a smooth $X'/S$. Does this sound fine?
Upd. So, smoothly embeddable is better? Is it ok to call a morphism smoothly ...
- 16.9k
6
votes
0
answers
224
views
If $X,Y$ are regular and of finite type over $S$, can $X\times _S Y$ be embedded into a regular $S$-scheme?
It seems to be well-known (see Is there an example of a variety over the complex numbers with no embedding into a smooth variety?) that a general finite type $S$-scheme does not embedd into a regular $...
- 16.9k
7
votes
0
answers
286
views
Level p characteristic 2 modular forms and thetas
BACKGROUND
Let p be an odd prime. An element of Z/2[[x]] is "modular of level p" if it is the mod 2 reduction of a g in Z[[x]] with g the Fourier expansion of a modular form for gamma_0(p). In ...
- 5,422
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
220
views
Dimension of faithful irreducible representations of $\mathbb{Z}_q\rtimes \mathbb{Z}_{p^2}$ in characteristic p,q
Let $p,q$ be primes s.t. $q=np+1$. Denote $m=p^2$. Then $\mathbb{Z}_p$ acts non-trivially on $\mathbb{Z}_q$, so we have a non-abelian semi-direct product $\mathbb{Z}_q\rtimes \mathbb{Z}_m$, with the ...
- 407
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
28
votes
3
answers
2k
views
Intuitive pictures in characteristic p
This is a tough one, but does anyone know of any images that recall characteristic p geometry (over algebraically closed fields) in some sense? It is not enough if it is some picture that can be also ...
- 2,215
0
votes
0
answers
103
views
Why do I get a morphism $f_P: Spec \mathcal{O}_K \to \mathcal{X}$ for every point $P\in X(K)$? How does this morphism look like?
Hello,
my question probably isn't too hard, but I can't find the answer.
Let $K$ be a number field, $\mathcal{X} /\mathcal{O}_K$ be an arithmetic surface, which is a regular model for a projective, ...
- 1
0
votes
0
answers
669
views
Quicker way to show that the restriction to a open subvariety is again proper?
Dear all,
Let $f: X \rightarrow Y$ be a morphism of projective varieties over $\mathbb{C}$. Also let $V \subset Y$ be a nontrivial open subvariety and set $U:= f^{-1}(V)$.
I would like to show that $...
- 479
5
votes
0
answers
432
views
Is the pushout of smooth varieties along a smooth closed subvariety again a variety?
The following question is motivated by a desire to find a rough analog in algebraic geometry of the usual notion of gluing of smooth bordisms.
Suppose k is an algebraically closed field of ...
- 37.8k
13
votes
1
answer
2k
views
Which local ringed spaces are schemes?
(This was originally asked on math.stackexchange, but didn't get any responses. I figured it might be worthwhile to move it here and try again.)
This paper gives a proof that the underlying ...
- 639
3
votes
1
answer
390
views
Torsion of elliptic curves is finite
Let $S$ be an integral 1-dimensional scheme with function field $K$.
Let $E$ be an elliptic curve over $K$. The torsion of $E$ over $K$ is not necessarily finite. As an example consider an elliptic ...
- 295
10
votes
1
answer
1k
views
How does descent theory imply a sheaf is a scheme?
I've noticed that often authors will comment that "descent theory" shows that some sheaf in the étale topology is actually a scheme. I was wondering what result in descent theory actually implies ...
- 2,108
0
votes
0
answers
440
views
Foliations in positive characteristic
Ekedahl wrote about foliations in positive characteristic, over the field $\mathbb{Z}/p\mathbb{Z}$ as a subsheaf of the tangent sheaf, that are closed with respect to involution and $p$-power.
My ...
- 1
15
votes
2
answers
611
views
Minimal number of generators for $A^n$
Let $A$ be a commutative ring and $n \in \mathbb{N}$. What is the minimal number $e_A(n)$ of generators of the $A$-algebra $A^n$? Here is what I already know (I can add proofs if necessary) from a ...
- 63.1k
6
votes
1
answer
650
views
Finitely-affine morphisms; cohomological dimension of schemes
Let $f\colon X\to U$ be a morphism of Noetherian schemes such that the scheme $U$ is affine and the scheme $X$ is separated and, e.g., quasi-projective over affine. Let $U=\bigcup_\alpha U_\alpha$ be ...
- 15.6k
7
votes
1
answer
2k
views
(Mixed) Tate motives
Hi there,
in recent times I was reading texts about motives, and I want to ask
something about Tate motives which is not clear to me (as I came across
different definitions in different texts).
Let ...
- 4,605
2
votes
0
answers
245
views
Segre class of cones and Base change of projective cones
I'm trying to work out a result in Fulton's intersection theory and I think I need the following basic result about base change of projective cones (whose support may not be the entire base scheme).
...
- 3,555
5
votes
0
answers
86
views
Category of the smooth formal p-groups over a local ring
Fontaine showed in Asterisque 47-48 that the category of finite dimensional smooth formal $p$-groups over the ring $A=W(k)$ of the Witt vectors over a finite field $k$ is anti-equivalent to the ...
- 51
1
vote
0
answers
192
views
"Higher" Tangent spaces in char-p geometry - definition?
Hi, everyone!
I have some construction that requires exact definition.
I considered polynomial homomorphisms from $\Bbbk$ to $U_n(\Bbbk)$ (unitriangular matrices), where $char \Bbbk = p>0$ (more ...
- 1,422
10
votes
0
answers
323
views
The mod 3 reduction of some powers of delta
Let f in Z/3[[x]] be the mod 3 reduction of the Fourier expansion of the normalized weight 12 cusp form delta for the full modular group. The exponents appearing in f are all 1 mod 3. Fix
k>0 and ...
- 5,422
8
votes
0
answers
1k
views
Two definitions of smoothness?
This is confusing, there appear to be possibly two definitions of smoothness in algebraic geometry for a morphism $f: X \rightarrow Y$ of schemes of finite type over an arbitrary field $k$.
...
- 3,555
14
votes
1
answer
1k
views
Frobenius splitting of Fano varieties
Dear MO,
Question 1. Do you know of an example of a Fano variety which is not Frobenius split?
Background
(1) A variety $X$ in characteristic $p$ is called Frobenius split if there is a "$p$-th ...
- 16.2k
2
votes
1
answer
495
views
Linear systems over non-algebraically closed field
Let $X$ be a regular, irreducible projective scheme, of finite type over an arbitrary field $k$. Both Weil divisors and Cartier divisors are defined on $X$ and naturally correspond. If $k$ is ...
- 3,555
15
votes
6
answers
3k
views
Characteristic zero and characteristic $p$ in algebraic geometry
Are there non-trivial (i.e. excluding concepts that can be defined only for $p>0$) statements in algebraic geometry that hold for all fields of characteristic $p$ for all prime $p$ but are known ...
user16974
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
0
votes
0
answers
539
views
Projective spaces with nonconstant regular functions
I can construct a scheme by patching that represents a projective space over an arbitrary ring. I can also prove that, if the ring is a Jacobson domain, the only regular functions on it are constants.
...
- 133
7
votes
3
answers
652
views
Irreducible "family" of relative effective divisors of a smooth morphism
Let $\pi:X\rightarrow Y$ be a smooth proper (assume projective if needed) morphism of schemes with $Y$ locally noetherian, and let $Z\subset X$ be an irreducible integral closed subscheme containing ...
- 1,811
7
votes
0
answers
207
views
Unicritical rational functions on curves in characteristic $p$
Let $k$ be an algebraically closed field of positive characteristic $p$, and let $X_{/k}$ be a smooth projective connected curve. Let $x_0$ be a point of $X(k)$.
How precisely can one describe the ...
- 1,199
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
4
votes
1
answer
918
views
Heisenberg group in characteristic two
I have seen two constructions called the Heisenberg group. If $k$ is a field of characteristic not equal to $2$ and $V$ is a $2n$-dimensional vector space over $k$ with symplectic form $\omega : V \...
- 3,605
4
votes
2
answers
1k
views
Is the absolute Galois group of the field of Laurent series in positive characteristic finitely generated?
If $K$ is an algebraically closed field of characteristic $p>0$, then $K((t))$, the field of Laurent series with coefficients in $K$, has infinitely many Galois extensions of degree $p$. Indeed, ...
- 3,362
1
vote
0
answers
238
views
Classification of fibres in pencils of curves of genus two
For the case of characteristic positive, there exist some clasification of families of curves of genus two over a elliptic curve?.
- 11
8
votes
2
answers
1k
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K-Theory of Schemes: Monoidal vs. Exact
There are several ways for defining the K-Theory of a category depending on which structure it admits. The K-Theory of schemes is commonly defined as the "group completion" of the category of ...
- 333