All Questions
22,548 questions
3
votes
0
answers
483
views
Questions about Shimura curves
1: Suppose $A_3 $ is the moduli space of abelian varieties of dimension 3 .Is the union of all one dimension shimura varieties in $A_3 $ connected?
2: Given a Shimura curve (explicit construction), ...
- 709
1
vote
0
answers
534
views
Integral element in the quotient of a polynomial ring
Hello,
I'm writting a "report" (to learn) on algebraic geometry, and was looking to write a proof for the following statement :
Theorem : Let $K$ be an algebraically closed field and $C_1, C_2 \...
- 11
7
votes
0
answers
538
views
Some sort of descent theory
Let $X$ be a smooth, projective curve defined over a finite field $k$. We denote by $\overline{X}$ it's extension to the algebraic closure $\overline{k}$. All the questions are about coherent sheaves.
...
- 887
2
votes
0
answers
64
views
morphism flat and relations
I want to know if the following result is true:
Let $I=<f_1,...,f_r>$ be an ideal of $K[x_1,x_2,...x_n,t_1,..,t_m]$. The morphism $\pi:V(I)\subset{K^{n+m}}\rightarrow{K^m}$, with $\pi(x,t)=t$, ...
- 21
1
vote
0
answers
231
views
Lower bound for intersection number
The base scheme is an algebraically closed field.
Let $X\to \mathbf{P}^1$ be an arithmetic surface over $\mathbf{P}^1$ and let $P$ be a section of $X\to \mathbf{P}^1$. Let $D$ be an effective (edited)...
- 225
3
votes
0
answers
281
views
What would be a characteristic-$p$ analogue for $C^{\infty}$-fiber bundles?
I'd like to know a notion for a morphism between algebraic varieties in characteristic $p$ that plays the role of a $C^{\infty}$-fiber bundle. It should be, in particular, flat. I'm not assuming the ...
- 4,265
3
votes
2
answers
378
views
Upper bound on greatest prime of bad reduction for a plane curve
Background
We are given a curve with integer coefficients. I want to make a suggestion in another question (Computationally bounding a curve's genus from below?) into a deterministic algorithm ...
- 4,593
2
votes
0
answers
597
views
Gauss manin connection under base change?
Consider for a proper smooth projective morphism f: X--> S-->spec(k), A is a regular k algebra of finite type , S=spec(A), k is a field .
we know that for$ R^n f_*(\Omega ^._{X/S})$ we have a Gauss ...
- 709
0
votes
0
answers
218
views
References needed for representation theory of certain unipotent algebraic groups in characteristic zero
Due to my preoccupation with characteristic p > 0 representation theory of algebraic groups, I am quite ignorant of some basic facts pertaining to their characteristic zero theory, mostly concerning ...
- 511
3
votes
0
answers
546
views
Determining polynomial coefficients correctly
I am working with a dot product of 2 unit vectors in R3 that are algebraic.
I am trying to recover their original format in a number field, but I am not sure of how to go about doing this. I do use ...
- 130
5
votes
0
answers
352
views
Why is Pic^0(C) of a curve C a variety?
Let $C$ be an abstract non-singular curve.
I'm having a hard time finding a reference for why $\text{Pic}^0(C)$ is a variety.
Any pointers towards a reference would be appreciated.
- 51
1
vote
0
answers
114
views
Divisors, factorisations of matrix valued functions, and the Lorentz group
How to construct a complex projective variety with several classes of non-intersecting divisors? How to keep the answer concrete and simple, so that explicit calculations can be done? And the problem -...
- 1,143
3
votes
0
answers
225
views
How can one check that two line bundles on $\overline{M}_{0,n}$ coincide?
Let $X$ be the Deligne-Mumford compactification of $\mathcal{M}_{0,n}$. Suppose I have two (big) line bundles $L$ and $L'$ on $X$ and that I want to show that they are the same element of $Pic(X)$. Of ...
- 3,779
2
votes
0
answers
143
views
compatible resolutions of singularities
Let $U$ and $V$ be complex vector spaces with an action of a finite group $G$. Denote by $P_{G,d}$ the space of $G$-equivariant polynomial maps $U\longrightarrow V$ with degree less than or equal to $...
- 483
1
vote
0
answers
157
views
Classification of Principal G-Bundles on 2-dimensional manifolds vs. elliptic curves
I've been recently studying the classification of principal G-bundles over elliptic curves. Specifically I've been using the paper by Friedman, Morgan and Witten "Principal G-Bundles over elliptic ...
- 500
0
votes
0
answers
240
views
Orbits of Infinite Grassmannian
"Any $L^+G$-orbit in Gr is known to be an orbit of the form Oλ for some λ ∈ $X_*(T)$, and Oλ = Oμ iff λ and μ are conjugate by W and the orbits form a finite stratification of each of the $Gr_i$."
...
4
votes
0
answers
178
views
İs the deligne lusztig variety corresponding to group of type G(2) computed?
I know that the deligne kusztig varieties corresponding to Suzuki group, Ree group and PGU-_2'(q) are explictly computed. Are there any result for the group G(2). Here 'result' means equation of ...
- 225
2
votes
1
answer
186
views
Behaviour of Primes under Regular Coefficient Extensions
Let $K\hookrightarrow K'$ be a regular extension of fields, and $K[x_{1},\cdots,x_{n}]\hookrightarrow K'[x_{1},\cdots,x_{n}]$ the corresponding ring extension. Does every prime ideal of the first ring ...
- 125
1
vote
2
answers
156
views
How to study the behavior of a particular function on a Vector Space.
Let, $V$ be a vector space over a field $K.$ Let, $T$ be a function from $V$ to $V$ such that
$T(kX) = kT(X)$ for all $k \in K$ and for all $X \in V$ and also
$T(k + X) = T(X)$ for all $k \in K$ ...
- 73
4
votes
1
answer
290
views
Predicting the behavior of the normalization without normalizing
I've been doing some very messy computations with normalizations of various surfaces, and I really want to not have to do them.
So my question is this: Let $S$ be a dimension $2$ integral scheme (you ...
- 5,929
1
vote
1
answer
394
views
Restriction of divisors to the generic fiber
Given a fibered surface $X \rightarrow C$, with generic fiber $Y$ and a vector bundle $E$ on $X$.
Then the first Chern class $c_1(E)$ is a divisor on $X$, so one can restrict this divisor to the ...
- 1,391
1
vote
1
answer
146
views
Is every nontrivial morphism already injective in this case?
I'm a little bit suprised at the moment, so i'll ask here if I see this wrong:
Given a sheaf of algebras $R$ ( e.g. maximal order or Azumaya) on a smooth projective scheme $X$ with generic point $p$. ...
- 1,391
0
votes
1
answer
79
views
Connection between the number of vertices and the number of lattice points of the integer hull of a polytope?
Is there a connections between the number of vertices and the number of lattice points of $P_I$, the integer hull of a polytope $P$? Which is usually more difficult to determine?
Or if I have a bound ...
2
votes
1
answer
276
views
The meaning of ${\infty}^{k}$.
When we say, that, say, a surface contains ${\infty}^{k}$ lines, do we mean that it contains a k-parameter family of lines? Do we assume that this family is parametrized by a $P^{k}$, say, or we use ...
- 71
2
votes
0
answers
176
views
Small Question about the construction of closed subscheme.
Let $\mathscr I_0=\mathscr J + (x_0x_2,x_1x_2,x_2^2,x_3x_2,x_4x_2)$. where $\mathscr J$ is the ideal sheaf of a rational normal quartic curve in $\mathbb P^3_{x_0,x_1,x_3,x_4}$.
Now, construct closed ...
- 337
1
vote
1
answer
311
views
A rational point in the scheme of pointed degree n rational functions [0912.2227]
The following question is related to "Remark 2.2" in Christophe Cazanave's paper "Algebraic homotopy classes of algebraic functions". I decided to add the arxiv article-id to the questions title to ...
- 1,273
2
votes
1
answer
428
views
equation for abelian varieties with a given polarization
Let $A$ be an abelian variety of dimension g and a polarization $L$ of type $(d_1,.....,d_g)$ (let alone the case $d_i=d_j,$ $\forall i, j$). What is the degree of the generators of the homogeneous ...
- 3,779
1
vote
0
answers
267
views
subset embedding gives trefoil knot [closed]
Let $X$ be a topological space and $E_n(X)$ the space of finite sets of cardinality $\leq n$.
It is a theorem of Bott that $E_3(S^1)=S^3$. What is the idea to show that
the embedding $S^1\...
- 11
3
votes
1
answer
249
views
GW invariants for varieties with negative first Chern class
Does there exist any theorem claiming that if a variety with negative first Chern class has no rational curves then every GW invariant is zero?
2
votes
0
answers
365
views
Are schematic fixed points of a torus action on an affinized twistor deformation flat?
This is a follow-up to some earlier questions about flatness of schematic fixed points of certain deformations. Since I could never come up with good enough hypotheses in those examples, let me try a ...
- 44.7k
7
votes
0
answers
385
views
Riemann-Roch as an index theorem [closed]
I am sorry to make this a new question. I would have liked to leave a comment, but I suppose I don't have enough rep for that....
So, in the accepted answer to this question I don't understand why in ...
- 191
6
votes
0
answers
242
views
Are Tate twists of t-positive motives positive with respect to the Voevodsky's homotopy t-structure?
Let $X$ be a Voevodsky's motif (over a perfect field) that belongs to the positive part of the homotopy $t$-structure (i.e. its cohomology as an object of $D^-(ShSmCor)$ is zero in negative degrees). ...
- 16.9k
5
votes
0
answers
600
views
Barsotti--Weil formula over separably closed fields
Let $S$ be a noetherian scheme and let $A$ be an abelian scheme on $S$ with dual $A^\vee$. The generalised Barsotti Weil formula states that there is a canonical and functorial (in $S$ and $A$) ...
- 4,015
1
vote
1
answer
213
views
flatness of a kernel
Hi,
let $A$ an abelian scheme over a curve $C$ and $n$ an integer greather than 3 coprime with the characteristic of the ground field. Do you know why the kernel of the multiplication by $n$ is flat ...
- 31
3
votes
0
answers
308
views
Endomorphisms of Jacobians of Hyperelliptic Curves taking Exceptional Divisors to Exceptional Divisors
I am trying to find an hyperelliptic curve (say of genus 2) together with an endomorphism $\phi$ of its Jacobian with the property that it sends Mumford reduced divisors of the exceptional form $P-\...
- 281
7
votes
1
answer
259
views
5
votes
0
answers
682
views
Analysis analogue of Orlov's theorem?
Mukai's theorem states that if $X$ is an abelian variety, and $\check{X}$ is the dual abelian variety, then the Fourier-Mukai transform corresponding to the Poincare line bundle on $X \times \check{X}$...
- 21k
3
votes
1
answer
457
views
Intuition for Nagata's altitude formula?
This is theorem 14.C on p.84 of Matsumura's commutative algebra.
Let $A$ be a noetherian domain, and let $B$ be a finitely generated overdomain of $A$. Let $P \in Spec(B)$ and $p = P \cap A$. Then ...
user709
5
votes
0
answers
413
views
Frobenius splitting of tangent bundles of flag varieties
BACKGROUND
Let $X$ be a variety over an algebraically closed field $k$ of positive characteristic $p$. Let $F : X \to X$ denote the absolute Frobenius morphism, i.e. the morphism that is the identity ...
- 3,637
3
votes
0
answers
358
views
Some non-trivial and explicit shape of Kähler cone?
It may be difficult to give some special and non-trivial examples of Kähler cones.The examples I know are the following:
for complex tori, the Kähler cone is just the set of positive hermitian ...
- 247
6
votes
0
answers
208
views
Poisson Ind-Varieties
I am looking for any places in the literature where an author has had occasion to consider Poisson structures on infinite-dimensional algebro-geometric objects, e.g. ind-varieties or proalgebraic ...
- 2,806
1
vote
0
answers
91
views
maximal separated quotient scheme [duplicate]
Let $X$ be a scheme, not necessarily separated. Is there a notion "its maximally separated quotient"? This means, a separated scheme $Y$ with a morphism $X \to Y$, such that it is initial in all ...
- 131
1
vote
0
answers
171
views
From special entry locus to general entry locus
Let $X\subset\mathbb{P}^N$ be a complex irreducible non-degenerate projective variety.
If
$$
p\in \mathrm{Sec}(X):=\overline{\bigcup_{x_1,x_2\in X\atop x_1\neq x_2}\langle x_1, x_2\rangle},
$$
the ...
- 1,159
2
votes
0
answers
220
views
Invariance of anti-plurigenus of Fano varieties with canonical singularities under small deformation?
I want to know the reference for the following theorem (Theorem 5.28. in Kollar-Mori's book "Birational geometry of algebraic varieties"):
Theorem Let $f:X \rightarrow S$ be a proper flat morphism of ...
- 909
0
votes
1
answer
225
views
Codimension of non-common condition is 2?
If we have n homogeneous polynomials (over algebraically closed field) $f_1\ldots , f_n$ on variables $x_0, \ldots , x_n$
$$
f_i(x_0, \ldots , x_n) = \sum_{j_0,\ldots , j_n} a_{i, j_0, \ldots , j_n} ...
- 1,422
1
vote
0
answers
204
views
Which rational subfields are corresponding to the two dimensional subspaces of holomorphic differentials
I implemented the algorithm that Felipe Voloch's suggested in his reply to the question:
Subfields of a function field
the algorithm is here:
Subfields of a function field
I considered the ...
- 601
1
vote
0
answers
276
views
What is known about the Picard scheme of a complete toric variety over C?
Let $X$ be a complete toric variety over a field $k$. Its Picard scheme is defined to be the scheme representing the functor $Pic_{(X/k)(\text{fppf})}$, where $Pic_{X/k}: Sch_k^{op}\to Set$ sends a ...
- 5,052
1
vote
0
answers
163
views
On explicit eigenfunctions
Given an algebraic surface $S$ defined by an algebraic equation such
as $x^{4}+2y^{4}+3z^{4}=1$, how would one find the third smallest
eigenvalue $\mu_{3}$ for the differential equation $\Delta f\left(...
- 11
0
votes
1
answer
261
views
Flat locus of $S_{1}$-morphism
Hi, everybody.
Consider an ${\rm S}_{1}$- morphism $f:X\rightarrow S$ of reduced complex spaces. Assume that $f$ is open (universally open in Alg.geom), equidimensional with $n$-pure dimensional ...
- 435
0
votes
0
answers
71
views
Compatibility of maps on points under base change
Let $S$ be an arbitrary scheme, and let $X,S'$ be $S$-schemes.
Using e.g. EGA I (3.3.9), (3.3.14), one obtains for any $S'$-scheme $T$, viewed as an $S$-scheme via $S'\to S$, a canonical bijection ...
