All Questions
61 questions
3
votes
1
answer
200
views
Square root of relative Kähler differentials and families of curves
Let $f: X \to S$ be a smooth morphism of schemes of relative dimension $1$. Then $K=\Omega^1_{X/S}$ is a line bundle on $X$. I am interested in the following question:
When does $\Omega_{X/S}$ have a ...
- 12.9k
3
votes
3
answers
1k
views
The non-existence of the fine moduli scheme of vector bundles. Why?
The reference I am using is Hoffmann - The moduli stack of vector bundles on a curve. The question is about the moduli space of vector bundles. I am trying to understand why the fine moduli scheme ...
- 3,429
1
vote
0
answers
116
views
Universal picard variety of degree d
Let $M_g$ denote the moduli space of smooth genus $g$ curves over $\mathbb{C}$, where $g \geq 2$. Let $Pic^{d,g}$ denote the universal picard variety over $M_g$, parameterizing pairs $(C,L)$ where $C$...
- 129
2
votes
1
answer
217
views
Dual of slope semistable vector bundle on higher dimensional variety
Recall the definition of slope semistability, taken from section 1.2 of Huybrechts and Lehn's "Geometry of Moduli Spaces of Sheaves" book. Let $X$ be a projective $\mathbb{C}$-scheme and $E \...
- 2,727
1
vote
0
answers
47
views
Homology groups of moduli of parabolic bundles with fixed determinant
I am looking for the Homology groups of the moduli space of stable parabolic bundles over a smooth projective curve with fixed determinant.
In particular, what is the second homology group of such ...
- 195
2
votes
1
answer
129
views
Isomorphism between $\operatorname{End}_0(E)$ and $\operatorname{End}_0(E')$ as Lie algebra bundles
This may be a stupid question.
I'm reading the paper "Automorphisms of moduli spaces of vector bundles over a curve" of Indranil Biswas, Tomas L. Gomez, V. Munoz (arXiv link). I have a ...
CommunityBot
- 1
2
votes
0
answers
143
views
Fundamental group of the moduli space of parabolic bundles with fixed determinant
I am looking for the fundamental group of the moduli space of parabolic bundles with fixed determinant over a smooth projective curve.
I know that the fundamental group of the moduli space of vector ...
- 195
1
vote
1
answer
190
views
Semistable pure dimension one sheaves of rank 1 and degree 0 on a singular curve
We are working on a problem about semistable pure dimension one sheaves of rank $1$ and degree $0$ on a singular curve $C$ (for example, the Kodaira fiber of type $I_2$, i.e. $C=C_1\cup C_2$ where $...
- 39.3k
2
votes
1
answer
288
views
Is there a stable reduction for a family of vector bundles?
I. General Question
Consider a one-parameter family of vector bundles $E_t$ on a smooth projective variety $X$ with fixed Chern character $v$. Suppose $E_t$ is Gieseker stable when $t\neq 0$ and $E_0$ ...
- 39.3k
3
votes
0
answers
125
views
Absolutely indecomposable objects and moduli space
In the setting of moduli spaces of vector bundles/quiver representations etc I've encountered a few times situations like the one that I'll try to explain thereafter .I was wondering if there's a &...
- 1,061
1
vote
0
answers
155
views
Codimension of the complement of the stable locus
Let $\mathcal{M}$ denote the moduli space of semistable vector bundles of fixed rank and degree over a compact Riemann surface $X$. Let $\mathcal{M}^s \subset \mathcal{M}$ be the moduli of stable ...
- 195
61
votes
11
answers
21k
views
What are some open problems in algebraic geometry?
What are the open big problems in algebraic geometry and vector bundles?
More specifically, I would like to know what are interesting problems related to moduli spaces of vector bundles over ...
- 19.6k
2
votes
1
answer
226
views
Picard group of moduli of principal bundles
I am looking for the Picard group of the moduli space of principal $G$-bundles for a connected reductive complex algebraic group $G$.
Is it isomorphic to $\mathbb{Z}$? If not, what can we say when $G=\...
- 8,529
4
votes
1
answer
392
views
$Ext$-algebra of stable vector bundles
Let $X$ be a smooth projective variety over $\mathbb{C}$ and $E$ a slope-stable vector bundle on $X$ with regard to some ample line bundle $H$.
Question: What can we say about the algebra structure of ...
- 7,300
5
votes
1
answer
418
views
Jumping conics in Grassmannians
Let $Gr(1,n)$ be the Grassmannian of lines in $\mathbb{P}^n$, and $f:\mathbb{P}^1\rightarrow Gr(1,n)$ a morphism of degree two. The pull-back $f^{*}S$ of the tautological bundle $S$ on $Gr(1,n)$ ...
- 39.3k
1
vote
0
answers
131
views
Schur's lemma for sheaves with different reduced Hilbert polynomials
Recall Schur's Lemma for Gieseker-semistable sheaves, in particular the injectivity statement:
Let $\psi : F \to G$ be a morphism of Gieseker-semistable sheaves. If $p(F)=p(G)$ and $F$ is stable, ...
- 123
4
votes
1
answer
281
views
Torelli theorem for stable vector bundle
Let $X, X^{\prime}\colon$ smooth projective curve on $\mathbb{C}$ (genus $\geq 3$),
$M(r,d)\colon$ coarse moduli of stable vector bundles with rank $r\geq2$, and degree $d$ , and
$M(r,\xi)\colon$ ...
- 148k
1
vote
0
answers
79
views
A question about Hitchin discriminant
Let $X$ be a smooth projective curve over $\mathbb{C}$, $\mathcal{M}$ be moduli space of Higgs bundle of rank $r\geq2$ and degree $d$ on $X$, $W=\bigoplus_{i=2}^{r}H^{0}(X,K_{X}^{\otimes i})$, and $H\...
- 297
1
vote
0
answers
89
views
The dimension of parameter space of unstable Higgs bundle
Let $X$:smooth projective curve of genus $g\geq 3$ over $\mathbb{C}$, $\mathcal{M}(r,d)$:moduli space of stable Higgs bundles of rank $r\geq 2$ and degree $d$ on $X$, and $N$:moduli space of stable ...
- 297
1
vote
0
answers
115
views
global algebraic functions $\Gamma(T^{*}M)$ on the cotangent bundle of moduli space
Let $X\colon$ smooth projective curve,
$\mathcal{M}\colon $ moduli space of semistable higgs bundle of rank $r$ and with fixed determinat $\xi$, and
$H\colon \mathcal{M}\rightarrow W=\oplus_{i=2}^{r} ...
- 297
2
votes
0
answers
110
views
The openness the set of $s\in \bigoplus H^0(C,K_{C}^{\otimes i})$ for which the spectral curve is irreducible and reduced
Let $C$ be smooth projective curve, $\mathcal{M}$ be the moduli space of semistable Higgs bundles, $H:\mathcal{M}\rightarrow W= \bigoplus H^0(C,K_{C}^{\otimes i})$ be the Hitchin map, and $\pi :C_s\...
- 297
4
votes
0
answers
129
views
Global algebraic function over the moduli space of semistable higgs bundles $\mathcal{M}$
Let $X$ be a smooth projective curve over $\mathbb{C}$, $\mathcal{M}$ be the moduli space of semistable higgs bundles, and $h: \mathcal{M}\rightarrow W=\bigoplus H^0(X,K_X^{\otimes{i}})$ be the ...
- 297
2
votes
0
answers
271
views
Fibers of Hitchin fibration are equidimensional
Let $X$ be a smooth projective curve over $\mathbb{C}$ of genus $g\ge 3$, $M$ be a moduli space of stable vector bundles on $X$ of rank $n\ge 2$ and degree $d$, $\mathcal{M}$ be a moduli space of ...
- 297
1
vote
0
answers
134
views
The compactified Jacobian is birational to a $\mathbb{P}^1$-fibration over the Jacobian of normalization
Let $Y$ be an integral curve whose only singularity is one simple node at a point $y$, and
$\pi:X\rightarrow Y$ be the normalization with $\pi^{-1}(y)=\{x,z\}$. $J(X)$ is the Jacobian of $X$, and $\...
- 297
1
vote
0
answers
253
views
Fiber of the Hitchin map
Let $C$ be smooth projective curve, $\mathcal{M}$ be the moduli space of semistable Higgs bundles, $H:\mathcal{M}\rightarrow W= \bigoplus H^0(C,K_{C}^{\otimes i})$ be the Hitchin map, and $\pi :C_s\...
- 297
0
votes
0
answers
411
views
Definition of Hitchin map
This may be a dumb question.
$\mathcal{M}(r,d)$ is a coarse moduli scheme for semistable pairs $(E,\phi:E \rightarrow K_X \otimes E)$ of rank $r$, degree $d$ on a smooth projective curve $X$ over $\...
- 297
5
votes
0
answers
173
views
Hitchin map and vector bundles
I've been learning a bit about automorphisms of moduli spaces of vector bundles and the Hitchin map.
I'm reading this paper of Indranil Biswas, Tomas L. Gomez, V. Munoz, and I have a problem about ...
- 297
1
vote
0
answers
360
views
On definition of stable vector/Higgs bundle
Recall that the slope of a holomorphic vector bundle $\mathcal{E}$ over a smooth projective variety (or rather a compact Kähler manifold) $X$ is defined as
$\mu(\mathcal{E}) :=\frac{\operatorname{deg}(...
- 1,170
3
votes
1
answer
402
views
Stable torsion free sheaf on smooth projective surface
Let $E$ be a torsion-free sheaf on a smooth projective variety $X$ over $\mathbb{C}$. Let $H$ be an ample line bundle on $X$. Then we say $E$ is stable if $\mu_{H}(F)<\mu_{H}(E),\,\forall 0\neq F \...
5
votes
1
answer
551
views
Relative logarithmic cotangent bundle
Let $\mathcal X \rightarrow S$ be a flat family of projective varieties over a discrete valuation ring $S$ such that the generic fibre $\mathcal X_{\eta}$ (say) is smooth projective variety and the ...
- 63.9k
3
votes
1
answer
325
views
vector bundles over projective line over an affine line
Let $k$ be a field and $E$ be a vector bundle over $\mathbb{P}_{k}^{1}\times\mathbb{A}_{k}^{1}$, does it extend to
$\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}$?
- 2,017
4
votes
0
answers
229
views
Is the determinant map $det:\mathcal{M}(r,d)\rightarrow Pic^d(X)$ on moduli space of semistable vector bundles a fibration?
Let $X$ smooth projective curve over $\mathbb{C}$, fix a line bundle $L$ of degree $d$, and let $\mathcal{M}(r,d)$ denote the moduli space of semistable vector bundles of rank $r$ and degree $d$. It ...
- 737
5
votes
1
answer
771
views
Confusion in known result about moduli space of vector bundle of rank 2 degree 0 vector bundles over smooth curve of genus 2
Theorem: Let $X$ be a complete, non-singular algebraic curve of genus
$2$. Let $U(2, \Theta)$ be the space of $S$-equivalence classes of semi-stable vector
bundles of rank $2$ and degree $\Theta$. The ...
- 3,486
3
votes
1
answer
326
views
Dimension of Moduli space of Parabolic $G$- Higgs bundles
I am looking at Usha Bhosle's paper on Moduli of Parabolic $G$-bundles. She calculated the dimension of the moduli space.
Now I want to know the the dimension of the moduli of parabolic $G$-Higgs ...
- 8,529
2
votes
1
answer
197
views
When is the moduli of generalized parabolic bundles with fixed determinant smooth?
Let $X$ be a smooth, projective curve of genus at least $2$, $x_1, x_2$ two distinct closed points, $d$ an odd integer and $\alpha$ a positive real number less than $1$. By a generalized parabolic ...
- 17.4k
3
votes
1
answer
960
views
Isomorphism classes of line bundles with connections
Isomorphism classes of line bundles over a scheme $X$ are described by the Picard group $Pic(X)$. Now there is a paper that describes the moduli space of line bundles with connections. This paper is ...
CommunityBot
- 1
0
votes
0
answers
171
views
Surjectivity of the Albanese map of the moduli space of stable vector bundles
I have a naive question:(I saw that it is related to relative K-theory of Hodge-Deligne and also Nadel-Chern-Weil theory )
Let $\mathcal M (r, d)$ be the moduli space of stable vector bundles of rank ...
- 189
3
votes
2
answers
708
views
On a result of Sir Michael Atiyah "Vector bundles on Elliptic curves" Theorem 2 Page 426
Let $V$ be a fixed vector space. Let $X$ be a smooth curve. Consider the Quot scheme $Q$ of Quotients of $V\otimes \mathcal{O}_X$ of degree $d$ and rank $r$. Let $R$ be the open subscheme of $Q$ ...
- 39.3k
3
votes
0
answers
356
views
Universal moduli space of rank two vector bundles over a curve
Let $\mathcal M_C$ denote the moduli space of rank two vector bundles over a smooth proper curve $C$ over an algebraically closed field $k$ of characteristic zero. Let $k(t)$ denote the function ...
- 18.7k
5
votes
0
answers
296
views
Bundles as Extensions and Jump Phenomena
Let $C$ be a Riemann Surface of genus $g \geq 2$. Consider a Vector Bundle of rank $r$ and degree $d$ on $C$. It is often convenient to construct such a Vector Bundle as an extension
\begin{equation}
...
- 1,073
9
votes
2
answers
4k
views
Reference request: moduli spaces of vector bundles
I am trying to study the moduli spaces of holomorphic vector bundles quickly, and I'm primarily interested in understanding:
Why and where the stability condition is used.
How are the moduli spaces ...
- 1,980
5
votes
1
answer
488
views
Isomorphism classes of sheaves which arise as extensions
Let $X$ be a proper(say, smooth) variety and $E,F$ are coherent sheaves on it. Extensions of $E$ by $F$ are parametrised by a finite-dimensional vector space $\mathrm{Ext}^1(E,F)$. I am intersted in ...
- 39.3k
8
votes
1
answer
635
views
Higgs bundles and stable vector bundle
Let $\mathcal M_X(r,0,K_X)$ be tha moduli space of semistable Higgs bundles over a smooth irreducible algebraic curve over $\mathbb C$. And let $E$ be a stable vector bundle of rank $r$ and degree $0$....
- 1,891
3
votes
0
answers
405
views
Some queries on Moduli Space of Parabolic Vector Bundles on curve
In "Moduli of Vector Bundles on curves with Parabolic Structures"-Bulletin of the American Mathematical Society Volume 83, Number 1, January 1977 the author announces the following result on moduli ...
- 26.3k
12
votes
2
answers
2k
views
Moduli space of (all) vector bundles on $\mathbb{P}^1$
It is well known that, by a theorem of Grothendieck, every vector bundle (always assumed coherent in this question; and everything is over the complex numbers) splits as a direct sum of line bundles.
...
CommunityBot
- 1
7
votes
1
answer
391
views
When is a general sheaf (on the projective plane) globally generated?
Let $v$ be a chern character on $\mathbb P^2$ so that the moduli of sheaves of chern character $v$ is non-empty of the expected dimension. When is it true that the general sheaf in moduli is globally ...
- 1,509
5
votes
0
answers
162
views
Flatness of Chern classes for flat family of sheaves
Let $Q$ be a quasi-projective $k$-scheme (not necessarily smooth), $X$ a smooth projective $k$-variety and $\mathcal E$ a family of (torsion free) sheaves on $X$ parametrized by $Q$. Suppose that $\...
- 59
4
votes
0
answers
302
views
Cohomology and deformations of moduli of vector bundles
I believe that the following is well-known, but I cannot find a reference
in the literature...
Let $X$ be a smooth variety (in our case $X = M^s(r)$ coarse moduli space of stable rank $r$ vector ...
- 3,779
4
votes
0
answers
304
views
Dimension of the singular locus of $\mathcal M_X(r,d)$
Let $X$ be an algebraic smooth curve of genus $g$ over $\mathbb C$, and let $\mathcal M_X(r,d)$ (resp $\mathcal M_X^0(r)$) be the moduli space of vector bundle of rank $r$ and degree $d$ (resp. with ...
- 1,891
1
vote
0
answers
122
views
Is it possible to find an explicit definition of the "universal" (co)tangent bundle?
Let $H_{0,1}(\mathbb{P}^2, d)$ be the space of holomorphic degree $d$
maps (that are not multiply covered) from $\mathbb{P}^1$ to $\mathbb{P}^2$ with one marked point
$y \in \mathbb{P^1} $ $\textit{...
- 3,245