All Questions
22,548 questions
0
votes
1
answer
278
views
Boundary operator in the colimit of complexes.
Let $\mathcal{A}$ be an abelian category and let $\mathcal{C}$ be the (abelian) category of complexes of objects in $\mathcal{A}$. Suppose we have a small indexing category $\mathcal{I}$ and a functor ...
0
votes
0
answers
261
views
Is an immersed Kronecker join always a multilinear variety on a Hilbert space?
The question asked is:
Is the implicitization of an arbitrary-rank immersed Kronecker join always a multilinear variety on a Hilbert space?
This is related to another MathOverflow question
In ...
- 1,389
0
votes
0
answers
212
views
On 'special properties' of various 'sheaf image' functors for a local complete intersection morphism
Let $f:X\to Y$ be a local complete intersection morphism (of schemes or varieties) of (relative) dimension $c$ everywhere. Is it true that $f^!\cong f^*[2c]$ (as a functor between the derived ...
- 16.9k
3
votes
0
answers
576
views
Does an étale equivalence relation of schemes induce an equivalence relation on points?
Let $R \rightrightarrows U \to X$ be a presentation of an algebraic space by schemes.
Does this induce an exact sequence $|R| \rightrightarrows |U| \to |X|$ on underlying points?
The reason I ask is ...
- 1,538
6
votes
0
answers
402
views
What is known about line bundles on the tangent bundle of a flag variety?
Let $G$ be a semisimple algebraic group over an algebraically closed field of arbitrary characteristic. (I'm most interested in the positive characteristic case). Let $B \subseteq G$ be a Borel ...
- 3,637
3
votes
0
answers
204
views
Computing Elliptic Curves of Conductor Divisible by a Large Prime Factor
A little while ago, I came across a paper (or slides from a talk or something) that seemed to suggest that the modular symbol method for computing elliptic curves over $\mathbf{Q}$ of prescribed ...
- 309
1
vote
0
answers
85
views
Complements to reducible plane projective curves
Hello,
Suppose that $C_1,C_2\subset\mathbb P^2$ are projective curves (over $\mathbb C$); $C_1$ and $C_2$ may be reducible but they must not have a common component. Let $L\subset \mathbb P^2$ be a ...
- 1,939
0
votes
0
answers
83
views
Analogue of Knudsen clutching
Is there an analogue of Knudsen clutching for moduli stacks of "pointed" varieties?
I admit this question is not very precise. I'm really asking two questions though.
Are there analogues of the ...
- 246
3
votes
0
answers
293
views
Polarizations on $M_{0,n}$ from Kapranov's quotient constructions
In Kapranov's marvelous paper Chow quotients of Grassmannian I, he proves that $\overline{M}_{0,n}$ is isomorphic to both the Hilbert quotient and Chow quotient $(\mathbb{P}^1)^n//\text{SL}_2$. These ...
- 1,871
7
votes
0
answers
320
views
Naturality of the associated sheaf
Let $X$ be a small site and $C$ a bicomplete (i.e. complete and cocomplete) category (or rather assume enough so that associated sheaves exists; see the comments). Consider the functor $\text{PSh}(X;C)...
- 63.1k
2
votes
0
answers
170
views
Intersections of General positions of system of linear subspaces
It should be a basic question, but somehow, I doubt if it is doable by elementary methods.
Fix a vector space $V$ over a field $k$ with $\dim V=r+1$. Consider the moduli space parametrizing $(V_1, ...
- 21
5
votes
1
answer
283
views
how good an approximation to the equivariant derived category is given by the Grassmannian filtration of the classifying space?
So, let's say one has an action of $GL_n$ on an algebraic variety $X$ over a field $k$, and two objects $F,G$ in the equivariant derived category (i.e., the derived category of constructible sheaves ...
- 44.7k
0
votes
1
answer
346
views
Algebraic Correspondences 'Expressible' as Vector Bundles
For algebraic curves $C$ over a closed field, a correspondence on $C$ is a the same thing as a divisor, and so, a line bundle on $C \times C$. Can I assume that this simplification does not extend to ...
- 3,409
5
votes
0
answers
202
views
Integral models for the moduli stack of vector bundles
Let $\mathcal{R}$ be a discrete valuation ring of unequal characteristic and $\mathcal{K}$ be its field of fractions. Let $X$ be a smooth proper curve over $Spec(K)$ and $\mathcal{X}$ be a smooth ...
- 491
3
votes
0
answers
273
views
What is the definition of an invariant of elliptic curve?
I know what is the $j$-invariant but I am asking about a general definition in sence of classical invariant theory.
The following possible definition seems to be wrong:
Consider an elliptic ...
- 301
2
votes
0
answers
314
views
The Artin stack of vector bundles\ cohrent sheaves on a surface.
Given $C$ a smooth, projective, algebraic curve, the Artin stack $Bun_{r,d}$ is smooth, irreducible (I think) of dimension $r^2(g-1)$. I am interested in the Artin stack $M_{r,c}$ of vector bundles of ...
- 773
5
votes
0
answers
238
views
When does the normalization have regular special fiber?
Let's say $\mathcal{O}$ is a complete DVR with fraction field $K$ and algebraically closed residue field $k$. (The case I had in mind here was with $\mathcal{O}$ of equicharacteristic $p$, so assume ...
- 4,487
2
votes
1
answer
254
views
Lie algebra actions on schemes
Let us assume first of all that we are in the affine case (we can worry about globalization later) and that we have $X$ affine over $S$, where $S$ is some unspecified scheme (but in practice probably ...
- 1,841
2
votes
0
answers
278
views
Euler characteristic of cyclic quotients
Suppose that $T$ is a smooth complex projective algebraic surface, such that the finite cyclic group
$\mathbb{Z_n}$ acts on it. Also, suppose that $S$ is another complex projective algebraic surface (...
- 395
2
votes
0
answers
86
views
Bisections in Kan Complexes
Kan Complexes can be seen as a generalization of groupoids, mostly called (weak)
infinity groupoids in this context.
On groupoids we can define the \textbf{group of bisections} the following way:
...
- 624
5
votes
0
answers
438
views
Primary decomposition for non-affine schemes
I will call a (nonzero) ring primary if every zero divisor is nilpotent. (This implies that the prime spectrum is irreducible, although the converse does not hold.) An irreducible scheme I will call ...
- 7,338
2
votes
1
answer
100
views
on a lower bound related to albanese map
As we know, the albanese map $Alb$ assoicates a smooth proper variety $X$ of dimension $n$ to an abelian variety $Alb(X)$ of dimension $g=H^0(X,\Omega^1_X)$. Another well known fact is the moduli ...
- 211
-1
votes
1
answer
333
views
Quantum cohomology of isomorphic Poisson varieties
This question is related with my previous one Quantum cohomology rings as invariants, but now, I want to ask a more concrete thing. If $X$ and $Y$ are Poisson varieties which are isomorphic (as a ...
- 1
7
votes
0
answers
554
views
Functorial point of view of spectrum (Looking for reference)
I think I should elaborate a bit. What I am asking is the definition of spectrum of a category as a stack in functor view of points.
In noncommutative algebraic geometry. We define spectrum of an ...
- 5,515
1
vote
0
answers
89
views
field of coefficients, automorphism, cohomology
Let $k \subset \mathbb{C}$ be a number field and $X$ a smooth algebraic variety over $k$. Asssume $X$ is equipped with an automorphism $h$ of order $n$. Then $h$ induces linear maps in singular ...
- 11
0
votes
1
answer
82
views
Deriving the rule for direct differentiating Bezier functions
I was reading http://www.idav.ucdavis.edu/education/CAGDNotes/Bernstein-Polynomials/Bernstein-Polynomials.html, the section on the derivative, which gives the derivative for a Bezier curve as:
$B_{k,...
7
votes
0
answers
491
views
Alterations of regular varieties
Let $X$ be a regular quasi-projective variety over a perfect field $k$. The existence of a "good compactification" of $X$, i.e. a regular projective variety $\bar{X}$ with an embedding $X\...
- 4,450
1
vote
0
answers
205
views
Space of sections
If S is a noetherian scheme and π : Z → X a morphism of S-schemes,
where X is proper over S and Z is quasi-projective over S, then the set-valued
contravariant functor $\Pi_{Z/X/S}$ on locally ...
- 1,811
4
votes
0
answers
193
views
Generalized linear systems
Let $X$ be an algebraic variety and let $Z\subset X$ be a subvariety. Let $[Z]$ be the class defined by $Z$ in the Chow group. Let $L(Z)$ be set of effective algebraic cycles on $X$ linearly ...
- 83
1
vote
0
answers
64
views
Unfolding subspace algebraic space
Let $ Y \subset X$ be a closed subspace of an algebraic space of finite type over $\mathbb{C}$. Let $p : Y' \rightarrow Y$ be a proper map of algebraic spaces. Artin proved that there exists a ...
- 7,310
0
votes
0
answers
214
views
How do I find non-linear sets that are invariant under a certain linear transformation?
I have an invertible linear transformation $T:F^k\to F^k$, where $F$ is a finite field and $k$ is a natural number.
It's easy to find the linear subspaces S that are invariant under T.
How do I find ...
- 179
2
votes
0
answers
148
views
Support of Tor over affinoid algebras
Suppose $k$ is a complete nonarchimedian field, $A$ is a $k$-affinoid algebra, and $M$ is a finitely presented $A$-module. Is the set
$\tau(M)= \left\{ x \in \mathrm{Sp}(A)\,\mathrm{with}\,\mathrm{...
- 13.1k
3
votes
0
answers
479
views
torsion freeness of tensor product
Hi.
Let $f:A\rightarrow B$ be a morphism of local noetherian rings, $M$ (resp. $N$) a $B$ (resp. $A$-)-module of finite type. We assume that $prof_{A}(M)\geq 2$ and $N$ is torsion free.
Then it is ...
- 435
0
votes
0
answers
67
views
open subset in constructible set of divisors
Let a smooth projective curve $X$ over $\mathbb{C}$.
Let a pair $(x, D)$ a pair xith a closed point $x$ and $D$ an effective divisor on $X$, such that $d_{x}:=m_{x}(D)\neq 0$.
Let $N=\deg (D)$ and $X^...
- 3,472
1
vote
0
answers
226
views
Does tensoring by a reflexive sheaf induce isomorphism between Ext groups?
Let $X$ be a normal projective variety which has only terminal singularities.
Let $\Omega^1_X$ be the Kahler differential sheaf on $X$ and $\omega_X$ be the dualizing sheaf on $X$. For a coherent ...
- 909
2
votes
0
answers
380
views
Geometric Inertia Action
Let $K$ be a finite extension of $Q_p$ and $K'/K$ a totally ramified Galois extension
with Galois group $G$. For $g\in G$ and any scheme $X$ over $O_{K'}$, write
$X_g$ for the base change of $X$ ...
- 1,609
1
vote
0
answers
351
views
Regularity and limits of smooth rational curves.
Fix integers $2 < d \leq n$.
Suppose that $T$ is a smooth complex curve with marked point $0 \in T$, and $X$ is a closed subscheme of $\mathbb{P}^n_T$, flat over $T$ such that each fiber has ...
- 1,990
5
votes
0
answers
377
views
Constructing spherical embeddings from colored fans
The gist: how does one produce a spherical embedding from a colored fan?
The setup: $G$ is a reductive group, $B$ a borel subgroup, and $G/H$ a spherical homogeneous space; that is, it has an open $B$...
- 3,968
12
votes
0
answers
440
views
K-Weil cohomology theories?
I don't know very much about this stuff, so I'm a bit afraid that I'm being naive or stupid, and I apologize if I am --- but it seems to me that Weil cohomology theories, or at least the standard ...
- 21k
2
votes
0
answers
254
views
Projectivized Normal Cone to Satake Compactification
Let $\mathcal{A}_g$ be the moduli space of principally polarized abelian varieties over $\mathbb{C}$.
There exists a compactification, the Satake compactification, which is minimal and has the ...
- 16k
5
votes
0
answers
547
views
Why should we consider D-module on flag variety of Lie algebra?
Why don't we stay at D-module on base affine space but go to study flag variety of Lie algebra?
I remembered there are nice papers of Bernstein-Gelfand-Gelfand and Gelfand-Kirillov discussing the ...
- 51
2
votes
1
answer
125
views
Determinant bundles of rank $2$ sheaves on $K3$ surfaces.
Let $M$ be a rank-$2$ vector bundle on a $K3$ surface $S$ such that $h^0(M)\geq 2$ and $h^2(M)=0$. Is it possible that $h^2(\det M)>0$? If yes, can you give me some examples?
- 773
1
vote
0
answers
532
views
Quotient of an affine variety by an infinite discrete group
This is a pretty basic question, so I'd be happy with either standard references or with explanations. Also, there's a good chance I'm confused about some things in the statement of the question, and ...
- 2,869
0
votes
0
answers
131
views
Do "recoil/rebound" left mutations exist (on a Del Pezzo surface)?
I have been studying exceptional sheaves and their mutations on Del Pezzo surfaces (specifically, on $\mathbb{P}^1 \times \mathbb{P}^1$). Given an exceptional pair $(E,F)$ of sheaves there are a ...
- 1
2
votes
1
answer
466
views
Semiclassical explanation of "Structured" spaces [closed]
We have our general notions of manifolds, schemes, et cetera, and other geometric "spaces", and we realize that a lot of these look like topological spaces with structure sheaves i.e. structured ...
2
votes
1
answer
356
views
k-th Chow Group and k-th graded part of K_0 ismorphic for DM-stacks?
If X is an algebraic scheme, K_0(X) has a filtration by taking the subgroups generated by coherent sheaves whose support as at most dimension k. The associated graded groups are the quotients, and ...
- 3,917
4
votes
0
answers
338
views
What to call the following variant of tame ramification
Suppose that $R \subseteq S$ is a generically separable extension of 1-dimensional normal domains (you can assume that $R$ is local if you'd like) of equal-characteristic $p > 0$ (for simplicity, ...
- 20.5k
1
vote
0
answers
80
views
smooth algebras and triviality of de Rham complex
Hi,
Let $R$ be a $\mathbb Q$-algebra and let $A$ be a smooth $R$-algebra. If $A$ is a polynomial algebra
$A = R[T_1,\dots,T_n]$, then it is easy to see that the natural map
$R \to \Omega^\bullet_{A/R}...
- 2,842
2
votes
0
answers
355
views
Boundedness of Hilbert polynomials of hypersurfaces
Let $(X,H)$ be a smooth polarized projective variety of dimension $n$.
If $Y \subset X$ is an irreducible hypersurface then its degree is $H^{n-1} \cdot Y$,
and its Hilbert polynomial is $p_Y(t) = ...
- 8,828
2
votes
0
answers
83
views
Largest subsets of quadrics consisting of "nonorthogonal" vectors
Assume we have an $A$-module $M$, and a quadratic form $q : M \to A$. Recall that it means that
1) $q (a m) = a^2 q (m)$ for all $a \in A$ and $m \in M$, and
2) $B_q (x, y) := q (x + y) - q (x) - q ...
- 311