All Questions
22,546 questions
2
votes
0
answers
125
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Generalization of Koszul cohomology for wedge product with a fixed q-form: literature and references?
Let $A$ be a commutative ring. For an odd positive integer $q$ and an integer $p$ in the range $1 \leq p \leq q$, consider the cochain complex:
$0 \to \Lambda^p(A^N) \to \Lambda^{p+q}(A^N) \to \cdots \...
- 21
2
votes
0
answers
134
views
Universal semistable curve
For the moduli of stable curve, we have the result by Deligne-Mumford-Knudsen that there's an isomorphism of moduli spaces
$$\Phi : \overline{\mathcal{X}}_{g,n} \xrightarrow{\sim} \overline{\mathcal{M}...
- 834
2
votes
1
answer
201
views
Section 3 of Atiyah's "On analytic surfaces with double points" — some questions
I have some questions about section 3 of Atiyah's "On analytic surfaces with double points," a short 9 page paper. Section 3 is all dedicated to proving lemma 4.
Near the end of section 3, ...
- 129
2
votes
0
answers
120
views
Analogs of Plücker relations in Clifford algebras, and Bott periodicity (?)
Classical Plücker relations can be viewed as conditions on coefficients of an element $x=\sum_Sc_Se_S$, $S=(i_1,...,i_k)$, $i_1<\cdots<i_k$, $\{i_1,...,i_k\}\subset\{1,...,n\}$ of an exterior ...
- 18.4k
3
votes
1
answer
366
views
Variants of Grothendieck section conjecture
Let $X$ be a smooth projective variety defined over a field $k$. We fix the following notations : $\overline{k}$ denotes the algebraic closure of the field $k$, $X_{\overline{k}}$ denotes the variety $...
- 443
3
votes
1
answer
138
views
Can the coefficients of a Taylor series be expressed as rational functions for an affine variety?
Let $k$ be an algebraically closed field and $V \subseteq k^n$ an affine variety corresponding to a prime ideal $P \subseteq k[t_1, \dots, t_n]$. For $x\in V$ let $O_x = \{p/q \mid p,q\in k[t_1, \dots,...
- 696
3
votes
2
answers
366
views
Rational divisors on Calabi–Yau threefolds
Following the construction of [2], consider the full subcategory $\mathcal{D}_0 \subset D^\flat(\operatorname{Coh} \omega_{\mathbb{P}^1})$ consisting of complexes whose cohomology objects are ...
- 317
1
vote
0
answers
108
views
L.c.i locus of Hilbert scheme of points on singular varieties
Let $X$ be an algebraic variety over $\mathbb{C}$. What can we say about the l.c.i. locus of $\text{Hilb}^n(X)$?
When $X$ is smooth, it is well-known that the l.c.i. locus of $\text{Hilb}^n(X)$ is ...
- 385
4
votes
0
answers
240
views
What do we do when $G$ doesn't have a Shimura variety?
Let $G$ be a reductive group. If one can associate to $G$ a Shimura datum $(G,X)$, then the étale cohomology of the associated Shimura variety $\operatorname{Sh}(G,X)$ is a strong tool for the ...
- 57
0
votes
0
answers
89
views
Extend algebraic morphism to a compactification with normal crossing boundary
Suppose $X$ and $Y$ are smooth algebraic variety over a char $0$ field $k$, and $f:X\to Y$ a morphism. I want to ask whether there exists compactifications $\bar X$ and $\bar Y$ such that $\bar X\...
- 785
1
vote
0
answers
92
views
Compactification of smooth varieties with normal crossing boundary
I see in this paper, page 46, the second sentence of 4.1, that every smooth variety over a characteristic $0$ field can be embedded into a proper smooth variety with normal crossing boundary, and the ...
- 785
4
votes
0
answers
173
views
Why are the Hodge filtrations on cohomology canonically bounded?
If $X$ is a complex projective variety of dimension $n$ then the de Rham cohomology $H^{k}(X,\mathbb Q)$ naturally has a mixed Hodge structure with an increasing weight filtration $W_\bullet$ and a ...
- 141
2
votes
1
answer
127
views
Changing the weight space for an eigenvariety
Let $G$ be an algebraic group (like $\operatorname{GL}_2$ or $\operatorname{GL}_2 \times\operatorname{GL}_2$ for example). Assume that there exists an eigenvariety $\mathcal{E}^G$ parameterizing ...
- 93
2
votes
1
answer
185
views
Number of rational points of a quotient of connected linear algebraic groups
Let $H$ be a closed connected subgroup of a connected linear algebraic group $G$ over an algebraically closed field of characteristic $p>0$, and let $\sigma=\sigma_q$ be the standard Frobenius ...
- 23
3
votes
1
answer
133
views
Is a simply connected locally 2-connected complex a union of spheres and planes?
Let $X$ be a (potentially infinite) 2-dimensional simplicial complex. Then each link at a vertex $x\in X$ is a graph.
Question. If $X$ is simply connected and each link is 2-connected (in the sense ...
- 13.6k
-5
votes
0
answers
126
views
Is a quiver variety a moduli stack of quiver representations?
As the title, I was just wondering is a quiver variety a moduli stack of quiver representations? I am familiar with quiver varieties but not that familiar with moduli stacks, so I was just wondering ...
- 133
3
votes
1
answer
151
views
Locally nilpotent derivations and triangularizability
If $ k $ is a field of characteristic zero and $ \delta \in T_{\mathbb{A}^{n}_{k}/k} $, then $ \delta $ is triangular if $ \delta = \sum_{i=2}^{n} f_{i}(x_{1},\dots,x_{i-1}) \frac{\partial}{\partial ...
- 912
4
votes
1
answer
245
views
Group action on affine variety induces faithful action on tangent space
I have a queestion about the proof of Lemma 2.2 from the paper arxiv 1105.3739:
Let $G$ be a group acting faithfully on an irreducible affine variety $X=\operatorname{Spec}(A)$ over $k= \Bbb C$. ...
- 6,038
2
votes
0
answers
97
views
An injective map in equivariant algebraic K-theory
Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$ and $\mathfrak{g}$ be its Lie algebra. Let $\mathcal{N}$ be the nilpotent cone of $\mathfrak{g}$ consists of all nilpotent ...
- 653
0
votes
0
answers
151
views
Compactification of the Jacobian of singular curves via parabolic modules
I would like to better understand a certain compactification of the Jacobian variety of a singular algebraic plane curve as described in Cook's Ph.D. 1993 thesis Local and Global Aspects
of the Module ...
- 1
2
votes
0
answers
138
views
Leray spectral sequence for étale homology
Let $X$, $Y$, $Z$ be quasi-projective varieties over an algebraically closed field $k$, $f: X \to Y$ and $g: Z \to X$ proper (even projective) maps with $f$ smooth, and $h: Z \to Y$ their composite. ...
- 658
5
votes
4
answers
1k
views
Stable points in GIT: geometric picture
Is there a geometric picture justifying why "stable points" in GIT (Geometric Invariant Theory) are actually called "stable"? Stable, with respect to which effect? (Here, I have ...
- 619
4
votes
0
answers
82
views
Classification of nilpotent orbits over local fields (for type ABCD via partitions )
Let $\mathfrak g$ be a simple Lie algebra over a char $0$ local field $F$ (e.g. $F=\mathbb R$ or $F=\mathbb Q_p$) with its adjoint group $G$. Let $\mathcal N \subseteq \mathfrak g$ be its nilpotent ...
- 6,622
1
vote
0
answers
75
views
Parametrized moduli spaces of semistable bundles by varying Kähler classes
Inspired by Liviu Nicolaescu's answer here, I was eager on trying to come up with examples of parameter spaces $\Lambda$, configuration spaces $\mathscr{C}$ and parametrized moduli spaces $\mathscr{M}$...
- 137
3
votes
1
answer
215
views
Geodesic flows and Killing fields
How well-known is the following result:
Let $(M,g)$ be a closed Riemannian manifold and define $D:C^\infty(M,Sym^mT^*M)\to C^\infty(M,Sym^{m+1}T^*M)$, $D\alpha:=\mathbb S(\nabla\alpha)$ where $Sym^m$ ...
3
votes
0
answers
166
views
Are motives of K3 surfaces of abelian type?
I refer to the article of van Geemen https://arxiv.org/pdf/math/9903146. What van Geemen calls the Kuga-Satake-Hodge conjecture suggests that for a K3 surface $X$ over $\mathbb{C}$, the summand $h^2(X)...
- 658
5
votes
0
answers
181
views
Deformations of cotangent bundles
Let $M$ be a smooth variety of even dimension over $\mathbb C$. I am interested in necessary or sufficient conditions such that $X$ is a deformation of a family of cotangent bundles.
In other words, ...
- 6,622
2
votes
1
answer
156
views
$\mathbb{C}^*$-action on moduli space of Higgs bundles
Let $M_{r,d}$ be the moduli space of semistable Higgs bundles of rank $r$ and degree $d$ over a compact Riemann surface. Over $M_{r,d}$ we have a $\mathbb{C}^*$-action $$t \cdot (E,\phi)=(E, t \phi). $...
- 1,061
0
votes
0
answers
85
views
Taking hyperplane section remains dominant
I was reading Kollar's book "Rational curves on algebraic varieties". This is from Chapter IV, proposition $1.3$. I don't understand the proof from $1.3.3$ to $1.3.1$, Page- 182. Suppose I ...
- 59
0
votes
0
answers
145
views
Bundles on stacks
We want to define what is a morphism of bundles over an algebraic stack. If $X$ is an algebraic stack and $V_n$ is the stack of rank $n$ vector bundles, a vector bundle on $X$ will be a morphism of ...
- 11
5
votes
1
answer
263
views
Central isogeny, Shimura varieties and exceptional cases
For a simple complex Lie algebra $\mathfrak g$, its weight lattice is not equal to the root lattice (i.e. the center of its simply connected form is a non-trivial finite group) iff $\mathfrak g$ is of ...
- 6,622
4
votes
1
answer
183
views
About the reduction type of the Kodaira symbol of elliptic curves defined over p-adic local fields
Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $E$ be an elliptic curve defined over $K$. Tate's algorithm can be used to compute the Kodaira symbol of the reduction type of $E$.
However, I ...
- 77
3
votes
0
answers
75
views
What exactly does it mean for the moduli space of stable sheaves to have a universal family étale locally?
It is known that in general, the moduli space $M$ of (Gieseker) stable sheaves do not have a universal family. Therefore, a map $f \colon S \to M$ does not necessarily induce a family of sheaves.
On ...
3
votes
0
answers
147
views
Tate conjecture for singular varieties in terms of intersection homology
In his book “Mixed motives and algebraic K-theory”, Jannsen generalizes the Tate conjecture to a potentially singular projective variety $X$ over a finitely generated field. The statement is the same ...
- 658
4
votes
0
answers
267
views
Is a pro-algebraic group over $\mathbb{Q}_p$ with Galois action the inverse limit of Galois-equivariant quotients?
Let $\mathcal{G}$ be a pro-algebraic group over $\mathbb{Q}_p$ with a continuous action of $G_K$ for a field $K$ (if $\mathcal{G}$ were an abelian unipotent group, this is precisely a $p$-adic Galois ...
- 15.4k
2
votes
3
answers
182
views
Stabilizers of the action of Levi on abelianization of nilpotent radical
$\DeclareMathOperator\Lie{Lie}$Let $G$ be a simple connected reductive group over $\mathbb C$. Consider a parabolic subgroup $P=MU$ of $G$, where $M$ is a Levi of $P$ and $U$ is the unipotent radical ...
- 6,622
2
votes
0
answers
165
views
Definition for "almost simple" linear algebraic groups
Proposition 2.18 from "Elementary abelian $p$-subgroups of algebraic groups" by R. Griess. used the term "simply connected almost simple linear algebraic group $G$" without ...
- 217
1
vote
1
answer
158
views
Comparison of special metrics on Riemann surfaces with the hyperbolic one
Let $X$ be a complex algebraic curve, or equivalently a compact Riemann surface, of genus $a\ge2$. Denote $\omega_1, \dots , \omega_a$ a basis of holomorphic differentials, and consider the Riemaniann ...
- 31
1
vote
0
answers
69
views
Descent of $G$-invariant formal system of parameters using GAGF
Let $R=(R,\mathfrak{m})$ be a comm local regular ring of char $\neq 2$ (ie $2 \neq 0$ in $R$) with maximal ideal $\mathfrak{m}$ of (Krull) dimension $2$, ie $R$ admits system of parameters $x,y \in \...
- 6,038
9
votes
1
answer
402
views
Conceptual understanding of the Néron–Severi group
I'm trying to understand the importance of the Néron–Severi group $\operatorname{NS}(X)$ when $X$ is, say a complex manifold. My background is in the analytic side so I'm much more familiar with line ...
- 137
1
vote
0
answers
81
views
Is every homogeneous line bundle pulled back from the quotient stack?
Let $G= \mathbb{G}_m^k$ act on a variety $X$.
Let $\mathcal{L}$ be a line bundle on $X$ and assume that for each $g \in G$ the pullback $g^\star \mathcal{L}$ is isomorphic to $\mathcal{L}$.
Does it ...
- 323
6
votes
0
answers
135
views
Reconstructing a scheme from its quotient stack
Let $X/S$ be a scheme over a base $S$, with a group action by a nice group $G/S$ (typically $G/S$ affine smooth).
Can we reconstruct $X$ from its quotient stack $[X/G]$?
It seems that we can expect $X$...
- 645
1
vote
0
answers
72
views
Component groups of stabilizers for linear representations
Let $G$ be a connected simple reductive group over $\mathbb C$. Let $V$ be a finite-dimensional complex representation of $G$.
Given a vector $v \in V$, it is natural to consider its stabilizer group $...
- 6,622
5
votes
1
answer
301
views
Operations on filtered / graded vector spaces via $\mathbb{A}^1 / \mathbb{G}_m$
A well known fact (e.g. Moulinos - The geometry of filtrations) is that one can describe the category of filtered vector spaces as $\operatorname{FilVect} \simeq \operatorname{QCoh}(\mathbb{A}^1/\...
- 834
14
votes
1
answer
565
views
What is the "schematic" point of view for regular polyhedra?
Last week, I read Wikipedia's article on Alexander Grothendieck. It lists his twelve greatest contributions to mathematics as accounted for in Grothendieck's own Récoltes et Semailles. The final item ...
1
vote
0
answers
88
views
Equivariant resolution of singularity making a pullback of a line bundle admit a root
I am considering the following situation. Let $X$ be an irreducible normal projective variety with an action of a linear algebraic group $H$, and we have a $H$-equivariant line bundle $L$ over $X$. We ...
- 323
1
vote
0
answers
182
views
"Reflex field" for $\mathbb H/\Gamma$ for $\Gamma$ non-congruence
Suppose $\Gamma$ is a non-congruence arithmetic subgroup of $PGL_2(\mathbb Z)$, and $\mathbb H$ is the upper half plane of $\mathbb C$. Then by Belyi's theorem we know $\mathbb H/\Gamma$ is an ...
- 785
1
vote
0
answers
99
views
Cohen-Macaulayness of the homogeneous coordinate ring of projective monomial curves
Let $A = \{a_0, a_1, \ldots, a_{n-1}\} \subset \mathbb{N}$ be a set of non-negative integers where we assume that $a_0 < \cdots < a_{n-1}$ and set $d := a_{n-1}$. For every $s \in \mathbb{N}$, ...
1
vote
1
answer
418
views
Uses of the Mukai vector
Let $X$ be say a smooth projective variety. For $\mathcal{E}^\bullet \in D^b(X)$ the so-called Mukai vector is defined as $$v(\mathcal{E}^\bullet) = \operatorname{ch}(\mathcal{E}^\bullet)\sqrt{\...
- 137
47
votes
10
answers
6k
views
Algebraic theorems with no known algebraic proofs
What are some good examples of algebraic theorems that have no known algebraic proofs?
A few I know concern classifications of (not necessarily associative) division algebras over $\mathbb{R}$: the ...