scholarly journals The Picard group of Brauer-Severi varieties

2018 ◽  
Vol 16 (1) ◽  
pp. 1196-1203
Author(s):  
Eslam Badr ◽  
Francesc Bars ◽  
Elisa Lorenzo García
Keyword(s):  
Picard Group ◽  
Plane Curves ◽  
Base Field ◽  
Picard Groups ◽  
Severi Varieties ◽  
Smooth Plane

AbstractIn this paper, we provide explicit generators for the Picard groups of cyclic Brauer-Severi varieties defined over the base field. In particular,we provide such generators for all Brauer-Severi surfaces. To produce these generators we use the theory of twists of smooth plane curves.

scholarly journals Automorphism groups and Picard groups of additive full subcategories

2010 ◽  
Vol 107 (1) ◽  
pp. 5 ◽  
Author(s):  
Naoya Hiramatsu ◽  
Yuji Yoshino
Keyword(s):  
Automorphism Group ◽  
Direct Product ◽  
Full Subcategory ◽  
Automorphism Groups ◽  
Commutative Rings ◽  
Picard Group ◽  
Picard Groups

We study category equivalences between additive full subcategories of module categories over commutative rings. And we are able to define the Picard group of additive full subcategories. The aim of this paper is to study the properties of the Picard groups and show that the automorphism group of an additive full subcategory is a semi-direct product of the Picard group with the group of algebra automorphisms of the ring.

scholarly journals The Brauer–Picard groups of fusion categories coming from the ADE subfactors

2018 ◽  
Vol 29 (05) ◽  
pp. 1850036 ◽  
Author(s):  
Cain Edie-Michell
Keyword(s):  
Picard Group ◽  
Picard Groups ◽  
Fusion Categories ◽  
Planar Algebra

We compute the group of Morita auto-equivalences of the even parts of the [Formula: see text] subfactors, and Galois conjugates. To achieve this, we study the braided auto-equivalences of the Drinfeld centers of these categories. We give planar algebra presentations for each of these Drinfeld centers, which we leverage to obtain information about the braided auto-equivalences of the corresponding categories. We also perform the same calculations for the fusion categories constructed from the full [Formula: see text] subfactors. Of particular interest, the even part of the [Formula: see text] subfactor is shown to have Brauer–Picard group [Formula: see text]. We develop combinatorial arguments to compute the underlying algebra objects of these invertible bimodules.

scholarly journals Lower bounds for Clifford indices in rank three

2010 ◽  
Vol 150 (1) ◽  
pp. 23-33 ◽  
Author(s):  
H. LANGE ◽  
P. E. NEWSTEAD
Keyword(s):  
Lower Bounds ◽  
Vector Bundles ◽  
Plane Curves ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Smooth Plane ◽  
Semistable Vector Bundles

AbstractClifford indices for semistable vector bundles on a smooth projective curve of genus at least 4 were defined in previous papers by the authors. In this paper, we establish lower bounds for the Clifford indices for rank 3 bundles. As a consequence we show that, on smooth plane curves of degree at least 10, there exist non-generated bundles of rank 3 computing one of the Clifford indices.

scholarly journals The kernel of the monodromy of the universal family of degree d smooth plane curves

2020 ◽  
pp. 1-15
Author(s):  
Reid Monroe Harris
Keyword(s):  
Plane Curve ◽  
Teichmüller Space ◽  
Mapping Class ◽  
Plane Curves ◽  
Teichmuller Space ◽  
Infinite Rank ◽  
Universal Family ◽  
Smooth Plane ◽  
Quartic Curves ◽  
Countably Infinite

We consider the parameter space [Formula: see text] of smooth plane curves of degree [Formula: see text]. The universal smooth plane curve of degree [Formula: see text] is a fiber bundle [Formula: see text] with fiber diffeomorphic to a surface [Formula: see text]. This bundle gives rise to a monodromy homomorphism [Formula: see text], where [Formula: see text] is the mapping class group of [Formula: see text]. The main result of this paper is that the kernel of [Formula: see text] is isomorphic to [Formula: see text], where [Formula: see text] is a free group of countably infinite rank. In the process of proving this theorem, we show that the complement [Formula: see text] of the hyperelliptic locus [Formula: see text] in Teichmüller space [Formula: see text] has the homotopy type of an infinite wedge of spheres. As a corollary, we obtain that the moduli space of plane quartic curves is aspherical. The proofs use results from the Weil–Petersson geometry of Teichmüller space together with results from algebraic geometry.

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