scholarly journals Spherical Hall Algebras of Weighted Projective Curves

2018 ◽  
Vol 2020 (15) ◽  
pp. 4721-4775
Author(s):  
Jyun-Ao Lin
Keyword(s):  
Vector Bundles ◽  
Characteristic Functions ◽  
Projective Curve ◽  
Hall Algebra ◽  
Smooth Projective Curve ◽  
Coherent Sheaves ◽  
Hall Algebras ◽  
Arbitrary Genus ◽  
Projective Curves ◽  
Fixed Rank

Abstract In this article, we deal with the structure of the spherical Hall algebra $\mathbf{U}$ of coherent sheaves with parabolic structures on a smooth projective curve $X$ of arbitrary genus $g$. We provide a shuffle-like presentation of the bundle part $\mathbf{U}^>$ and show the existence of generic spherical Hall algebra of genus $g$. We also prove that the algebra $\mathbf{U}$ contains the characteristic functions on all the Harder–Narasimhan strata. These results together imply Schiffmann’s theorem on the existence of Kac polynomials for parabolic vector bundles of fixed rank and multi-degree over $X$. On the other hand, the shuffle structure we obtain is new and we make links to the representations of quantum affine algebras of type $A$.

scholarly journals ON THE VOEVODSKY MOTIVE OF THE MODULI STACK OF VECTOR BUNDLES ON A CURVE

2020 ◽  
Author(s):  
Victoria Hoskins ◽  
Simon Pepin Lehalleur
Keyword(s):  
Vector Bundles ◽  
Intersection Theory ◽  
Line Bundles ◽  
Projective Curve ◽  
Classifying Space ◽  
Smooth Projective Curve ◽  
Quot Schemes ◽  
Homotopy Colimit ◽  
Moduli Stack ◽  
Fixed Rank

Abstract We define and study the motive of the moduli stack of vector bundles of fixed rank and degree over a smooth projective curve in Voevodsky’s category of motives. We prove that this motive can be written as a homotopy colimit of motives of smooth projective Quot schemes of torsion quotients of sums of line bundles on the curve. When working with rational coefficients, we prove that the motive of the stack of bundles lies in the localizing tensor subcategory generated by the motive of the curve, using Białynicki-Birula decompositions of these Quot schemes. We conjecture a formula for the motive of this stack, inspired by the work of Atiyah and Bott on the topology of the classifying space of the gauge group, and we prove this conjecture modulo a conjecture on the intersection theory of the Quot schemes.

scholarly journals A Torelli type theorem for nodal curves

2021 ◽  
pp. 2150041
Author(s):  
Suratno Basu ◽  
Sourav Das
Keyword(s):  
Moduli Space ◽  
Vector Bundles ◽  
Type Theorem ◽  
Projective Curve ◽  
Fixed Degree ◽  
Smooth Projective Curve ◽  
Nodal Curve ◽  
Normal Crossing ◽  
Moduli Of Vector Bundles ◽  
Fixed Rank

The moduli space of Gieseker vector bundles is a compactification of moduli of vector bundles on a nodal curve. This moduli space has only normal-crossing singularities and it provides flat degeneration of the moduli of vector bundles over a smooth projective curve. We prove a Torelli type theorem for a nodal curve using the moduli space of stable Gieseker vector bundles of fixed rank (strictly greater than [Formula: see text]) and fixed degree such that rank and degree are co-prime.

Maximal Subbundles of Rank 2 Vector Bundles on Projective Curves

2000 ◽  
Vol 43 (2) ◽  
pp. 129-137 ◽  
Author(s):  
E. Ballico
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Stable Rank ◽  
Maximal Degree ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Stable Vector Bundles ◽  
Projective Curves ◽  
Rank 2 ◽  
Rank 2 Vector Bundles

AbstractLet E be a stable rank 2 vector bundle on a smooth projective curve X and V(E) be the set of all rank 1 subbundles of E with maximal degree. Here we study the varieties (non-emptyness, irreducibility and dimension) of all rank 2 stable vector bundles, E, on X with fixed deg(E) and deg(L), L ∈ V(E) and such that .

scholarly journals A pathological case of the C1 conjecture in mixed characteristic

2018 ◽  
Vol 167 (01) ◽  
pp. 61-64 ◽  
Author(s):  
INDER KAUR
Keyword(s):  
Line Bundle ◽  
Moduli Space ◽  
Projective Curve ◽  
Free Sheaf ◽  
Smooth Projective Curve ◽  
Mixed Characteristic ◽  
Locally Free Sheaf ◽  
Fixed Rank ◽  
Pathological Case

AbstractLet K be a field of characteristic 0. Fix integers r, d coprime with r ⩾ 2. Let XK be a smooth, projective, geometrically connected curve of genus g ⩾ 2 defined over K. Assume there exists a line bundle ${\cal L}_K$ on XK of degree d. In this paper we prove the existence of a stable locally free sheaf on XK with rank r and determinant ${\cal L}_K$. This trivially proves the C1 conjecture in mixed characteristic for the moduli space of stable locally free sheaves of fixed rank and determinant over a smooth, projective curve.

scholarly journals Moduli spaces of orthogonal and symplectic bundles over an algebraic curve

2008 ◽  
Vol 144 (3) ◽  
pp. 721-733 ◽  
Author(s):  
Olivier Serman
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Algebraic Curve ◽  
Vector Bundles ◽  
Explicit Description ◽  
Representation Space ◽  
Classical Groups ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Polynomial Invariants

AbstractWe prove that, given a smooth projective curve C of genus g≥2, the forgetful morphism $\mathcal {M}_{\mathbf {O}_r} \longrightarrow \mathcal {M}_{\mathbf {GL}_r}$ (respectively $\mathcal M_{\mathbf {Sp}_{2r}}\longrightarrow \mathcal M_{\mathbf {GL}_{2r}}$) from the moduli space of orthogonal (respectively symplectic) bundles to the moduli space of all vector bundles over C is an embedding. Our proof relies on an explicit description of a set of generators for the polynomial invariants on the representation space of a quiver under the action of a product of classical groups.

On the Cohomology of Moduli of Vector Bundles and the Tamagawa Number of SLn

2006 ◽  
Vol 58 (5) ◽  
pp. 1000-1025 ◽  
Author(s):  
Ajneet Dhillon
Keyword(s):  
Vector Bundles ◽  
Hodge Structure ◽  
Betti Numbers ◽  
Mixed Hodge Structure ◽  
Projective Curve ◽  
Ground Field ◽  
Smooth Projective Curve ◽  
Poincaré Polynomial ◽  
Moduli Stack ◽  
Moduli Of Vector Bundles

AbstractWe compute some Hodge and Betti numbers of the moduli space of stable rank r, degree d vector bundles on a smooth projective curve. We do not assume r and d are coprime. In the process we equip the cohomology of an arbitrary algebraic stack with a functorial mixed Hodge structure. This Hodge structure is computed in the case of the moduli stack of rank r, degree d vector bundles on a curve. Our methods also yield a formula for the Poincaré polynomial of the moduli stack that is valid over any ground field. In the last section we use the previous sections to give a proof that the Tamagawa number of SLn is one.

scholarly journals On the Rationality of Certain Strata of the Lange Stratification of Stable Vector Bundles on Curves

2001 ◽  
Vol 8 (4) ◽  
pp. 665-668
Author(s):  
E. Ballico
Keyword(s):  
Vector Bundles ◽  
Maximal Degree ◽  
Projective Curve ◽  
Fiber Structure ◽  
Generic Fiber ◽  
Smooth Projective Curve ◽  
Stable Vector Bundles ◽  
Natural Way

Abstract Let 𝑋 be a smooth projective curve of genus 𝑔 ≥ 2 and 𝑆(𝑟, 𝑑) the moduli scheme of all rank 𝑟 stable vector bundles of degree 𝑑 on 𝑋. Fix an integer 𝑘 with 0 < 𝑘 < 𝑟. H. Lange introduced a natural stratification of 𝑆(𝑟, 𝑑) using the degree of a rank 𝑘 subbundle of any 𝐸 ∈ 𝑆(𝑟, 𝑑) with maximal degree. Every non-dense stratum, say 𝑊(𝑘, 𝑟 – 𝑘, 𝑎, 𝑑 – 𝑎), has in a natural way a fiber structure ℎ : 𝑊(𝑘, 𝑟 – 𝑘, 𝑎, 𝑑 – 𝑎) → Pic𝑎(𝑋) × Pic𝑏(𝑋) with ℎ dominant. Here we study the rationality or the unirationality of the generic fiber of ℎ.

ON MAXIMALLY FROBENIUS DESTABILISED VECTOR BUNDLES

2019 ◽  
Vol 99 (2) ◽  
pp. 195-202
Author(s):  
LINGGUANG LI
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Closed Field ◽  
Projective Curve ◽  
Stable Vector Bundle ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field

Let $X$ be a smooth projective curve of genus $g\geq 2$ over an algebraically closed field $k$ of characteristic $p>0$. We show that for any integers $r$ and $d$ with $0<r<p$, there exists a maximally Frobenius destabilised stable vector bundle of rank $r$ and degree $d$ on $X$ if and only if $r\mid d$.

scholarly journals EQUATIONS OF THE MODULI OF HIGGS PAIRS AND INFINITE GRASSMANNIAN

2009 ◽  
Vol 20 (08) ◽  
pp. 1029-1055 ◽  
Author(s):  
D. HERNÁNDEZ-SERRANO ◽  
J. M. MUÑOZ PORRAS ◽  
F. J. PLAZA MARTÍN
Keyword(s):  
Fixed Point ◽  
Moduli Space ◽  
Vector Bundles ◽  
Higgs Field ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Spectral Cover ◽  
Relative Version ◽  
Moduli Of Vector Bundles

In this paper the moduli space of Higgs pairs over a fixed smooth projective curve with extra formal data is defined and is endowed with a scheme structure. We introduce a relative version of the Krichever map using a fibration of Sato Grassmannians and show that this map is injective. This, together with the characterization of the points of the image of the Krichever map, allows us to prove that this moduli space is a closed subscheme of the product of the moduli of vector bundles (with formal extra data) and a formal anologue of the Hitchin base. This characterization also provides us with a method for explicitly computing KP-type equations that describe the moduli space of Higgs pairs. Finally, for the case where the spectral cover is totally ramified at a fixed point of the curve, these equations are given in terms of the characteristic coefficients of the Higgs field.

Moduli of Rank 2 Stable Bundles and Hecke Curves

2016 ◽  
Vol 59 (4) ◽  
pp. 865-877
Author(s):  
Sarbeswar Pal
Keyword(s):  
Moduli Space ◽  
Vector Bundles ◽  
Short Note ◽  
Equivalence Classes ◽  
Complex Numbers ◽  
Stable Bundles ◽  
Smooth Projective Curve ◽  
Stable Vector Bundles ◽  
Arbitrary Genus ◽  
Rank 2

AbstractLet X be a smooth projective curve of arbitrary genus g > 3 over the complex numbers. In this short note we will show that the moduli space of rank 2 stable vector bundles with determinant isomorphic to Lx , where Lx denotes the line bundle corresponding to a point x ∊ X, is isomorphic to a certain variety of lines in the moduli space of S-equivalence classes of semistable bundles of rank 2 with trivial determinant.

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