All Questions
61 questions
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 ...
14
votes
0
answers
2k
views
conformal blocks for beginners
I have given now a couple of talks that involve conformal blocks bundles on the moduli stack $\overline{\mathcal{M}}_{g,n}$, in front of a public of algebraic geometers but not specialists of the ...
- 3,779
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.
...
- 23.3k
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 ...
9
votes
6
answers
3k
views
Reference request: Moduli spaces of bundles over singular curves
I would like to know some reference (articles, books...) about any kind of moduli spaces of any of the following objects:
vector bundles
torsion-free sheaves
principal bundles
parabolic bundles
over ...
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
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
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 $Gr(1,n)$ ...
- 8,998
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 ...
user149914
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 ...
- 7,377
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 ...
- 137
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
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
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
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
$M(r,\xi)\colon$ ...
- 297
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 ...
- 175
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
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
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
4
votes
0
answers
486
views
Stack of vector bundles (on a curve) over a strictly semi-stable point of the moduli space
Consider the stack $Bun_{r,d}^{ss}$ of rank $r$ semi-stable vector bundle of degree $d$ over a fixed curve. There exists also a coarse moduli space $M$ built via GIT. Over the stable locus of $M$ it ...
- 3,779
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 ...
- 587
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 ...
- 6,622
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
$\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}$?
- 3,472
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 ...
- 195
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 $Q$ ...
user100841
3
votes
1
answer
317
views
deformations of vector bundles on curves
Let $X$ be a smooth algebraic curve. Suppose I have a flat family $V_y\to X$ of vector bundles on $X$ over an affine scheme $S$. Let $p=Spec(k)$ be one geometric point of $S$. If the determinant of $...
- 3,779
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 \...
- 197
3
votes
1
answer
610
views
moduli of vector bundles on a surface
Let $S$ be a smooth projective surface with an ample divisor $X\subset S$. Consider the
moduli stack of vector bundles $F$ on $S$ such that
1) $c_1(F)=0$
2) $c_2(F)=n$
3) The restriction of $F$ to $...
- 7,037
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 ...
- 587
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
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
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 ...
- 290
2
votes
2
answers
998
views
Is the moduli space of stable vector bundles over a smooth projective curve fano?
Let $K$ be a field of characteristic zero but not algebraically closed. Let $C$ be a smooth projective curve over $K$. Let $r, d$ be two positive integers that are coprime. Consider the moduli space ...
- 1,593
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 \...
- 129
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 $E_0$ ...
- 1,803
2
votes
1
answer
765
views
fano moduli varieties of vector bundles
Let $M$ be a fine moduli space of vector bundles on curve which is an algebraic variety as well. The first example of such an object that I have in mind is rank 2, deg 1 VB on a genus 2 curve. This is ...
- 3,779
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=\...
- 195
2
votes
1
answer
168
views
stable vector bundle and space surves
I am sure this is well known, but I am not an expert...so I appreciate any help
Let $C \subset \mathbb{P}^3$ be the complete intersection of two hypersurfaces of degree $d_1$ and $d_2$. Let $U_{d_1,...
- 21
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 ...
- 297
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 ...
- 1,593
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
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
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
2
votes
0
answers
151
views
Moduli space of sheaves on a ribbon
In the paper "A non-linear deformation of the Hitchin dinamycal system", Donagi-Ein-Lazarsfeld describe the irreducible components of the moduli space $\mathcal M_R$ of stable sheaves of numerical ...
- 453
1
vote
2
answers
913
views
Connections on the Hodge bundle?
Let $\mathcal{M}_g$ be the moduli space of curves of genus $g$. Consider the holomorphic bundle $\mathcal{H}^k\rightarrow\mathcal{M}_g$ whose fiber over a curve $C\in\mathcal{M}_g$ is the space of ...
- 1,804
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 $...
- 11
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
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
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