Divisor Theory

2014 ◽  
pp. 43-55
Keyword(s):  
Divisor Theory

scholarly journals Some Theorems On Matrices With Real Quaternion Elements

1955 ◽  
Vol 7 ◽  
pp. 191-201 ◽  
Author(s):  
N. A. Wiegmann
Keyword(s):  
Similarity Transformation ◽  
Elementary Divisor ◽  
Quaternion Matrix ◽  
Unitary Similarity ◽  
Triangular Form ◽  
Quaternion Matrices ◽  
Divisor Theory ◽  
Real Quaternion ◽  
Unitary Similarity Transformation

Matrices with real quaternion elements have been dealt with in earlier papers by Wolf (10) and Lee (4). In the former, an elementary divisor theory was developed for such matrices by using an isomorphism between n×n real quaternion matrices and 2n×2n matrices with complex elements. In the latter, further results were obtained (including, mainly, the transforming of a quaternion matrix into a triangular form under a unitary similarity transformation) by using a different isomorphism.

The nullity of a wild knot in a compact 3-manifold

1973 ◽  
Vol 16 (3) ◽  
pp. 262-271
Author(s):  
James M. McPherson
Keyword(s):  
Fundamental Group ◽  
Present Author ◽  
Elementary Divisor ◽  
Divisor Theory ◽  
Infinitely Generated Modules ◽  
Torsion Free

The nullity of the Alexander module of the fundamental group of the complement of a knot in S3 was one of the invariants of wild knot type defined and investigated by E. J. Brody in [1], in which he developed a generalised elementary divisor theory applicable to infinitely generated modules over a unique factorisation domain. Brody asked whether the nullity of a knot with one wild point was bounded above by its enclosure genus; for knots in S3, the present author showed in [6] that this was indeed the case. In [7], it was (prematurely) stated by the author that this was also the case for knots k embedded in a 3-manifold M so that H,(M — k) was torsion-free.

scholarly journals Brill-Noether theory for curves of a fixed gonality

2021 ◽  
Vol 9 ◽  
Author(s):  
David Jensen ◽  
Dhruv Ranganathan
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Closed Formula ◽  
Noether Theorem ◽  
Stable Maps ◽  
Technical Result ◽  
General Curve ◽  
Divisor Theory ◽  
Arbitrary Genus ◽  
A Chain

Abstract We prove a generalisation of the Brill-Noether theorem for the variety of special divisors $W^r_d(C)$ on a general curve C of prescribed gonality. Our main theorem gives a closed formula for the dimension of $W^r_d(C)$ . We build on previous work of Pflueger, who used an analysis of the tropical divisor theory of special chains of cycles to give upper bounds on the dimensions of Brill-Noether varieties on such curves. We prove his conjecture, that this upper bound is achieved for a general curve. Our methods introduce logarithmic stable maps as a systematic tool in Brill-Noether theory. A precise relation between the divisor theory on chains of cycles and the corresponding tropical maps theory is exploited to prove new regeneration theorems for linear series with negative Brill-Noether number. The strategy involves blending an analysis of obstruction theories for logarithmic stable maps with the geometry of Berkovich curves. To show the utility of these methods, we provide a short new derivation of lifting for special divisors on a chain of cycles with generic edge lengths, proved using different techniques by Cartwright, Jensen, and Payne. A crucial technical result is a new realisability theorem for tropical stable maps in obstructed geometries, generalising a well-known theorem of Speyer on genus $1$ curves to arbitrary genus.

A note on subsemigroups with divisor theory

1995 ◽  
Vol 119 (3) ◽  
pp. 217-221 ◽  
Author(s):  
Krzystof Kuzara
Keyword(s):  
Divisor Theory
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