scholarly journals Vector bundles with a fixed determinant on an irreducible nodal curve

2005 ◽  
Vol 115 (4) ◽  
pp. 445-451 ◽  
Author(s):  
Usha N. Bhosle
Keyword(s):  
Vector Bundles ◽  
Nodal Curve

scholarly journals EXTENSIONS AND RANK-2 VECTOR BUNDLES ON IRREDUCIBLE NODAL CURVES

2005 ◽  
Vol 16 (10) ◽  
pp. 1081-1118
Author(s):  
D. ARCARA
Keyword(s):  
Moduli Space ◽  
Vector Bundles ◽  
Rational Map ◽  
Nodal Curves ◽  
Nodal Curve ◽  
Stable Vector Bundles ◽  
Rank 2 ◽  
Rank 2 Vector Bundles

We generalize Bertram's work on rank two vector bundles to an irreducible projective nodal curve C. We use the natural rational map [Formula: see text] defined by [Formula: see text] to study a compactification [Formula: see text] of the moduli space [Formula: see text] of semi-stable vector bundles of rank 2 and determinant L on C. In particular, we resolve the indeterminancy of ϕL in the case deg L = 3,4 via a sequence of three blow-ups with smooth centers.

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.

Fourier–Mukai Transform on a Compactified Jacobian

2018 ◽  
Vol 2020 (13) ◽  
pp. 3991-4015
Author(s):  
Usha N Bhosle ◽  
Sanjay Kumar Singh
Keyword(s):  
Moduli Space ◽  
Vector Bundles ◽  
Chern Classes ◽  
Nodal Curve ◽  
Injective Morphism ◽  
Compactified Jacobian ◽  
Fixed Rank

Abstract We use Fourier–Mukai transform to compute the cohomology of the Picard bundles on the compactified Jacobian of an integral nodal curve $Y$. We prove that the transform gives an injective morphism from the moduli space of vector bundles of rank $r \ge 2$ and degree $d$ ($d$ sufficiently large) on $Y$ to the moduli space of vector bundles of a fixed rank and fixed Chern classes on the compactified Jacobian of $Y$. We show that this morphism induces a morphism from the moduli space of vector bundles of rank $r \ge 2$ and a fixed determinant of degree $d$ on $Y$ to the moduli space of vector bundles of a fixed rank with a fixed determinant and fixed Chern classes on the compactified Jacobian of $Y$.

scholarly journals Nodal curves and polarizations with good properties

2021 ◽  
Author(s):  
Sonia Brivio ◽  
Filippo F. Favale
Keyword(s):  
Vector Bundles ◽  
Compact Type ◽  
Nodal Curves ◽  
Smooth Curves ◽  
Nodal Curve ◽  
Good For

AbstractIn this paper we deal with polarizations on a nodal curve C with smooth components. Our aim is to study and characterize a class of polarizations, which we call “good”, for which depth one sheaves on C reflect some properties that hold for vector bundles on smooth curves. We will concentrate, in particular, on the relation between the $${{\underline{w}}}$$ w ̲ -stability of $${\mathcal {O}}_C$$ O C and the goodness of $${{\underline{w}}}$$ w ̲ . We prove that these two concepts agree when C is of compact type and we conjecture that the same should hold for all nodal curves.

Banach Lie Algebroids

2011 ◽  
Vol 57 (2) ◽  
pp. 409-416
Author(s):  
Mihai Anastasiei
Keyword(s):  
Differential Equations ◽  
Smooth Manifold ◽  
Vector Bundles ◽  
Second Order ◽  
Lie Algebroids ◽  
Second Order Differential Equations

Banach Lie AlgebroidsFirst, we extend the notion of second order differential equations (SODE) on a smooth manifold to anchored Banach vector bundles. Then we define the Banach Lie algebroids as Lie algebroids structures modeled on anchored Banach vector bundles and prove that they form a category.

scholarly journals Fano 3-folds from homogeneous vector bundles over Grassmannians

2021 ◽  
Author(s):  
Lorenzo De Biase ◽  
Enrico Fatighenti ◽  
Fabio Tanturri
Keyword(s):  
Vector Bundles ◽  
Homogeneous Vector

AbstractWe rework the Mori–Mukai classification of Fano 3-folds, by describing each of the 105 families via biregular models as zero loci of general global sections of homogeneous vector bundles over products of Grassmannians.

scholarly journals EULER CLASSES: SIX-FUNCTORS FORMALISM, DUALITIES, INTEGRALITY AND LINEAR SUBSPACES OF COMPLETE INTERSECTIONS

2021 ◽  
pp. 1-66
Author(s):  
Tom Bachmann ◽  
Kirsten Wickelgren
Keyword(s):  
Commutative Algebra ◽  
Vector Bundles ◽  
Euler Class ◽  
Complete Intersections ◽  
Bilinear Forms ◽  
Euler Numbers ◽  
Coherent Sheaves ◽  
Linear Subspaces ◽  
Algebraic Vector

Abstract We equate various Euler classes of algebraic vector bundles, including those of [12] and one suggested by M. J. Hopkins, A. Raksit, and J.-P. Serre. We establish integrality results for this Euler class and give formulas for local indices at isolated zeros, both in terms of the six-functors formalism of coherent sheaves and as an explicit recipe in the commutative algebra of Scheja and Storch. As an application, we compute the Euler classes enriched in bilinear forms associated to arithmetic counts of d-planes on complete intersections in $\mathbb P^n$ in terms of topological Euler numbers over $\mathbb {R}$ and $\mathbb {C}$ .

scholarly journals Higher rank Clifford indices of curves on a K3 surface

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Soheyla Feyzbakhsh ◽  
Chunyi Li
Keyword(s):  
Vector Bundles ◽  
Plane Curve ◽  
Smooth Curve ◽  
Line Bundles ◽  
K3 Surface ◽  
Clifford Index ◽  
Smooth Plane ◽  
Higher Rank ◽  
Rank Vector ◽  
Semistable Vector Bundle

AbstractLet (X, H) be a polarized K3 surface with $$\mathrm {Pic}(X) = \mathbb {Z}H$$ Pic ( X ) = Z H , and let $$C\in |H|$$ C ∈ | H | be a smooth curve of genus g. We give an upper bound on the dimension of global sections of a semistable vector bundle on C. This allows us to compute the higher rank Clifford indices of C with high genus. In particular, when $$g\ge r^2\ge 4$$ g ≥ r 2 ≥ 4 , the rank r Clifford index of C can be computed by the restriction of Lazarsfeld–Mukai bundles on X corresponding to line bundles on the curve C. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher rank vector bundles. We also apply the same method to the projective plane and show that the rank r Clifford index of a degree $$d(\ge 5)$$ d ( ≥ 5 ) smooth plane curve is $$d-4$$ d - 4 , which is the same as the Clifford index of the curve.

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