scholarly journals Maximally Frobenius-destabilized vector bundles over smooth algebraic curves

2017 ◽  
Vol 28 (02) ◽  
pp. 1750003 ◽  
Author(s):  
Yifei Zhao
Keyword(s):  
Line Bundle ◽  
Vector Bundles ◽  
Algebraic Curves ◽  
Frobenius Morphism ◽  
Rank 2 ◽  
Characteristic 3

Vector bundles in positive characteristics have a tendency to be destabilized after pulling back by the Frobenius morphism. In this paper, we closely examine vector bundles over curves that are, in an appropriate sense, maximally destabilized by the Frobenius morphism. Then we prove that such bundles of rank 2 exist over any curve in characteristic 3, and are unique up to twisting by a line bundle. We also give an application of such bundles to the study of ample vector bundles, which is valid in all characteristics.

scholarly journals THE HYPERELLIPTIC THETA MAP AND OSCULATING PROJECTIONS

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.

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 Brill-Noether theory for vector bundles of rank $2$

1991 ◽  
Vol 43 (1) ◽  
pp. 123-126 ◽  
Author(s):  
Montserrat Teixidor i Bigas
Keyword(s):  
Vector Bundles ◽  
Rank 2

scholarly journals ADJUNCTION FOR VARIETIES WITH A ℂ* ACTION

2020 ◽  
Author(s):  
ELEONORA A. ROMANO ◽  
JAROSŁAW A. WIŚNIEWSKI
Keyword(s):  
Fixed Point ◽  
Line Bundle ◽  
Normal Bundle ◽  
Ample Line Bundle ◽  
Projective Manifold ◽  
Fano Manifolds ◽  
Adjunction Theory ◽  
Rank 2 ◽  
General Orbit ◽  
Complex Projective

Abstract Let X be a complex projective manifold, L an ample line bundle on X, and assume that we have a ℂ* action on (X;L). We classify such triples (X; L;ℂ*) for which the closure of a general orbit of the ℂ* action is of degree ≤ 3 with respect to L and, in addition, the source and the sink of the action are isolated fixed points, and the ℂ* action on the normal bundle of every fixed point component has weights ±1. We treat this situation by relating it to the classical adjunction theory. As an application, we prove that contact Fano manifolds of dimension 11 and 13 are homogeneous if their group of automorphisms is reductive of rank ≥ 2.

scholarly journals Quantum reconstruction for Fano bundles on projective space

2015 ◽  
Vol 218 ◽  
pp. 1-28
Author(s):  
Andrew Strangeway
Keyword(s):  
Projective Space ◽  
Vector Bundles ◽  
Quantum Cohomology ◽  
Lefschetz Theorem ◽  
Small Quantum ◽  
Low Degree ◽  
Reconstruction Theorem ◽  
Rank 2

AbstractWe present a reconstruction theorem for Fano vector bundles on projective space which recovers the small quantum cohomology for the projectivization of the bundle from a small number of low-degree Gromov-Witten invariants. We provide an extended example in which we calculate the quantum cohomology of a certain Fano 9-fold and deduce from this, using the quantum Lefschetz theorem, the quantum period sequence for a Fano 3-fold of Picard rank 2 and degree 24. This example is new, and is important for the Fanosearch program.

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