COHERENT SYSTEMS AND MODULAR SUBAVRIETIES OF $\mathcal{SU}_C(r)$

Let C be an algebraic smooth complex curve of genus g > 1. The object of this paper is the study of the birational structure of certain moduli spaces of vector bundles and of coherent systems on C and the comparison of different type of notions of stability arising in moduli theory. Notably we show that in certain cases these moduli spaces are birationally equivalent to fibrations over simple projective varieties, whose fibers are GIT quotients (ℙr-1)rg// PGL (r), where r is the rank of the considered vector bundles. This allows us to compare different definitions of (semi-)stability (slope stability, α-stability, GIT stability) for vector bundles, coherent systems and point sets, and derive relations between them. In certain cases of vector bundles of low rank when C has small genus, our construction produces families of classical modular varieties contained in the Coble hypersurfaces.

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LINEAR SERIES ON CURVES: STABILITY AND CLIFFORD INDEX

International Journal of Mathematics ◽

10.1142/s0129167x12501212 ◽

2012 ◽

Vol 23 (12) ◽

pp. 1250121 ◽

Cited By ~ 8

Author(s):

ERNESTO C. MISTRETTA ◽

LIDIA STOPPINO

Keyword(s):

Linear Stability ◽

Vector Bundles ◽

Linear Series ◽

Clifford Index ◽

Complex Curve ◽

Stable Vector Bundles ◽

Smooth Complex

We study concepts of stability associated to a smooth complex curve together with a linear series on it. In particular we investigate the relation between stability of the associated dual span bundle and linear stability. Our results imply that stability of the dual span holds under a hypothesis related to the Clifford index of the curve. Furthermore, in some of the cases, we prove that a stronger stability holds: cohomological stability. Finally, using our results we obtain stable vector bundles of slope 3, and prove that they admit theta-divisors.

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RATIONAL FAMILIES OF VECTOR BUNDLES ON CURVES

International Journal of Mathematics ◽

10.1142/s0129167x0400220x ◽

2004 ◽

Vol 15 (01) ◽

pp. 13-45 ◽

Cited By ~ 7

Author(s):

ANA-MARIA CASTRAVET

Keyword(s):

Moduli Space ◽

Moduli Spaces ◽

Vector Bundles ◽

Rational Curves ◽

Complex Curve ◽

Stable Vector Bundles ◽

Irreducible Components ◽

Rationally Connected ◽

Rank 2 ◽

Rank 2 Vector Bundles

Let C be a smooth projective complex curve of genus g≥2 and let M be the moduli space of rank 2, stable vector bundles on C, with fixed determinant of degree 1. For any k≥1, we find all the irreducible components of the space of rational curves on M, of degree k. In particular, we find the maximal rationally connected fibrations of these components. We prove that there is a one-to-one correspondence between moduli spaces of rational curves on M and moduli spaces of rank 2 vector bundles on ℙ1×C.

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Moduli Spaces of Vector Bundles over a Real Curve: ℤ/2-Betti Numbers

Canadian Journal of Mathematics ◽

10.4153/cjm-2013-049-1 ◽

2014 ◽

Vol 66 (5) ◽

pp. 961-992 ◽

Cited By ~ 7

Author(s):

Thomas Baird

Keyword(s):

Riemann Surface ◽

Moduli Space ◽

Moduli Spaces ◽

Vector Bundles ◽

Lagrangian Submanifolds ◽

Betti Numbers ◽

Stable Bundles ◽

Complex Curve ◽

Yang Mills ◽

Real Curve

AbstractModuli spaces of real bundles over a real curve arise naturally as Lagrangian submanifolds of the moduli space of semi–stable bundles over a complex curve. In this paper, we adapt the methods of Atiyah–Bott's “Yang–Mills over a Riemann Surface” to compute ℤ/2–Betti numbers of these spaces.

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Non-reduced moduli spaces of sheaves on multiple curves

Advances in Geometry ◽

10.1515/advgeom-2019-0033 ◽

2020 ◽

Vol 20 (2) ◽

pp. 285-296

Author(s):

Jean-Marc Drézet

Keyword(s):

Moduli Spaces ◽

Vector Bundles ◽

Versal Deformation ◽

Projective Varieties ◽

Coherent Sheaves ◽

Moduli Spaces Of Sheaves ◽

Projective Curves ◽

Locally Free Sheaf ◽

Stable Sheaves ◽

Rank 2 Vector Bundles

AbstractSome coherent sheaves on projective varieties have a non-reduced versal deformation space; for example, this is the case for most unstable rank 2 vector bundles on ℙ2, see [18]. In particular, some moduli spaces of stable sheaves are non-reduced. We consider some sheaves on ribbons (double structures on smooth projective curves): let E be a quasi locally free sheaf of rigid type and let 𝓔 be a flat family of sheaves containing E. We find that 𝓔 is a reduced deformation of E when some canonical family associated to 𝓔 is also flat. We consider also a deformation of the ribbon to reduced projective curves with two components, and find that E can be deformed in two distinct ways to sheaves on the reduced curves. In particular some components M of the moduli spaces of stable sheaves deform to two components of the moduli spaces of sheaves on the reduced curves, and M appears as the “limit” of varieties with two components, whence the non-reduced structure of M.

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NOTES ON COHERENT SYSTEMS

Revista de Matemática Teoría y Aplicaciones ◽

10.15517/rmta.v28i1.42154 ◽

2020 ◽

Vol 28 (1) ◽

pp. 1-38

Author(s):

ALEXANDER H.W. SCHMITT

Keyword(s):

Moduli Spaces ◽

Invariant Theory ◽

Vector Bundles ◽

Geometric Invariant Theory ◽

Geometric Invariant ◽

Coherent Systems ◽

Alternative Approach

We present an alternative approach to semistability and moduli spaces for coherent systems associated with decorated vector bundles. In this approach, it seems possible to construct a Hitchin map. We relate some examples to classical problems from geometric invariant theory.

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THE HYPERELLIPTIC THETA MAP AND OSCULATING PROJECTIONS

Nagoya Mathematical Journal ◽

10.1017/nmj.2020.37 ◽

2020 ◽

pp. 1-23

Author(s):

MICHELE BOLOGNESI ◽

NÉSTOR FERNÁNDEZ VARGAS

Keyword(s):

Moduli Space ◽

Moduli Spaces ◽

Vector Bundles ◽

Hyperelliptic Curve ◽

Geometric Description ◽

Birational Model ◽

Rank 2 ◽

Semistable Vector Bundles ◽

Ramification Locus

Abstract Let C be a hyperelliptic curve of genus $g \geq 3$ . In this paper, we give a new geometric description of the theta map for moduli spaces of rank 2 semistable vector bundles on C with trivial determinant. In order to do this, we describe a fibration of (a birational model of) the moduli space, whose fibers are GIT quotients $(\mathbb {P}^1)^{2g}//\text {PGL(2)}$ . Then, we identify the restriction of the theta map to these GIT quotients with some explicit degree 2 osculating projection. As a corollary of this construction, we obtain a birational inclusion of a fibration in Kummer $(g-1)$ -varieties over $\mathbb {P}^g$ inside the ramification locus of the theta map.

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