scholarly journals BRILL–NOETHER THEOREMS AND GLOBALLY GENERATED VECTOR BUNDLES ON HIRZEBRUCH SURFACES

2018 ◽  
Vol 238 ◽  
pp. 1-36 ◽  
Author(s):  
IZZET COSKUN ◽  
JACK HUIZENGA
Keyword(s):  
Projective Plane ◽  
Euler Characteristic ◽  
Chern Class ◽  
Vector Bundles ◽  
Betti Numbers ◽  
Stable Bundle ◽  
Hirzebruch Surface ◽  
Hirzebruch Surfaces ◽  
Chern Characters ◽  
First Chern Class

In this paper, we show that the cohomology of a general stable bundle on a Hirzebruch surface is determined by the Euler characteristic provided that the first Chern class satisfies necessary intersection conditions. More generally, we compute the Betti numbers of a general stable bundle. We also show that a general stable bundle on a Hirzebruch surface has a special resolution generalizing the Gaeta resolution on the projective plane. As a consequence of these results, we classify Chern characters such that the general stable bundle is globally generated.

Nef vector bundles on a quadric surface with the first Chern class (2, 1)

2020 ◽  
Vol 20 (1) ◽  
pp. 109-116
Author(s):  
Masahiro Ohno
Keyword(s):  
Chern Class ◽  
Characteristic Zero ◽  
Vector Bundles ◽  
Closed Field ◽  
Quadric Surface ◽  
Algebraically Closed Field ◽  
First Chern Class ◽  
Smooth Quadric Surface ◽  
Nef Vector Bundles

AbstractWe classify nef vector bundles on a smooth quadric surface with the first Chern class (2, 1) over an algebraically closed field of characteristic zero; we see in particular that such nef bundles are globally generated.

scholarly journals On the positivity of the first Chern class of an Ulrich vector bundle

2021 ◽  
pp. 2150071
Author(s):  
Angelo Felice Lopez
Keyword(s):  
Vector Bundle ◽  
Chern Class ◽  
Vector Bundles ◽  
First Chern Class

We study the positivity of the first Chern class of a rank [Formula: see text] Ulrich vector bundle [Formula: see text] on a smooth [Formula: see text]-dimensional variety [Formula: see text]. We prove that [Formula: see text] is very positive on every subvariety not contained in the union of lines in [Formula: see text]. In particular, if [Formula: see text] is not covered by lines we have that [Formula: see text] is big and [Formula: see text]. Moreover we classify rank [Formula: see text] Ulrich vector bundles [Formula: see text] with [Formula: see text] on surfaces and with [Formula: see text] or [Formula: see text] on threefolds (with some exceptions).

scholarly journals Nef Cones of Nested Hilbert Schemes of Points on Surfaces

2018 ◽  
Vol 2020 (11) ◽  
pp. 3260-3294
Author(s):  
Tim Ryan ◽  
Ruijie Yang
Keyword(s):  
Projective Plane ◽  
Birational Geometry ◽  
K3 Surface ◽  
Hilbert Schemes ◽  
Hirzebruch Surface ◽  
Hirzebruch Surfaces ◽  
Nef Cone

Abstract Let X be the projective plane, a Hirzebruch surface, or a general K3 surface. In this paper, we study the birational geometry of various nested Hilbert schemes of points parameterizing pairs of zero-dimensional subschemes on X. We calculate the nef cone for two types of nested Hilbert schemes. As an application, we recover a theorem of Butler on syzygies on Hirzebruch surfaces.

scholarly journals Stable vector bundles on an algebraic surface

1975 ◽  
Vol 58 ◽  
pp. 25-68 ◽  
Author(s):  
Masaki Maruyama
Keyword(s):  
Vector Bundle ◽  
Chern Class ◽  
Algebraic Curve ◽  
Algebraic Surface ◽  
Vector Bundles ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Stable Vector Bundles ◽  
First Chern Class

Let X be a non-singular projective algebraic curve over an algebraically closed field k. D. Mumford introduced the notion of stable vector bundles on X as follows;DEFINITION ([7]). A vector bundle E on X is stable if and only if for any non-trivial quotient bundle F of E,where deg ( • ) denotes the degree of the first Chern class of a vector bundles and r( • ) denotes the rank of a vector bundle.

scholarly journals The Chern character of the Verlinde bundle over ℳ¯ g,n

2017 ◽  
Vol 2017 (732) ◽  
pp. 147-163 ◽  
Author(s):  
Alina Marian ◽  
Dragos Oprea ◽  
Rahul Pandharipande ◽  
Aaron Pixton ◽  
Dimitri Zvonkine
Keyword(s):  
Explicit Formula ◽  
Chern Class ◽  
Chern Character ◽  
Conformal Blocks ◽  
Verlinde Formula ◽  
Fusion Algebra ◽  
Chern Characters ◽  
First Chern Class ◽  
Projective Flatness

Abstract We prove an explicit formula for the total Chern character of the Verlinde bundle of conformal blocks over \overline{\mathcal{M}}_{g,n} in terms of tautological classes. The Chern characters of the Verlinde bundles define a semisimple CohFT (the ranks, given by the Verlinde formula, determine a semisimple fusion algebra). According to Teleman’s classification of semisimple CohFTs, there exists an element of Givental’s group transforming the fusion algebra into the CohFT. We determine the element using the first Chern class of the Verlinde bundle on the interior {\mathcal{M}}_{g,n} and the projective flatness of the Hitchin connection.

scholarly journals GAUGE-EQUIVARIANT HILBERT BIMODULES AND CROSSED PRODUCTS BY ENDOMORPHISMS

2009 ◽  
Vol 20 (11) ◽  
pp. 1363-1396 ◽  
Author(s):  
EZIO VASSELLI
Keyword(s):  
Vector Bundle ◽  
Moduli Space ◽  
Chern Class ◽  
Vector Bundles ◽  
Crossed Products ◽  
Tensor Categories ◽  
Dual Object ◽  
Nontrivial Center ◽  
First Chern Class ◽  
The Given

C*-endomorphisms arising from superselection structures with nontrivial center define a 'rank' and a 'first Chern class'. Crossed products by such endomorphisms involve the Cuntz–Pimsner algebra of a vector bundle having the above-mentioned rank, first Chern class and can be used to construct a duality for abstract (nonsymmetric) tensor categories versus group bundles acting on (nonsymmetric) Hilbert bimodules. Existence and unicity of the dual object (i.e. the 'gauge' group bundle) are not ensured: we give a description of this phenomenon in terms of a certain moduli space associated with the given endomorphism. The above-mentioned Hilbert bimodules are noncommutative analogs of gauge-equivariant vector bundles in the sense of Nistor–Troitsky.

scholarly journals The Kodaira Problem for Kähler Spaces with Vanishing First Chern Class

2021 ◽  
Vol 9 ◽  
Author(s):  
Patrick Graf ◽  
Martin Schwald
Keyword(s):  
Chern Class ◽  
Cohomology Group ◽  
Canonical Bundle ◽  
Deformation Space ◽  
Projective Varieties ◽  
Quotient Singularities ◽  
Finite Quotient ◽  
First Chern Class ◽  
Small Deformations

Abstract Let X be a normal compact Kähler space with klt singularities and torsion canonical bundle. We show that X admits arbitrarily small deformations that are projective varieties if its locally trivial deformation space is smooth. We then prove that this unobstructedness assumption holds in at least three cases: if X has toroidal singularities, if X has finite quotient singularities and if the cohomology group ${\mathrm {H}^{2} \!\left ( X, {\mathscr {T}_{X}} \right )}$ vanishes.

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