Computing the Abel map

2008 ◽  
Vol 237 (24) ◽  
pp. 3214-3232 ◽  
Author(s):  
Bernard Deconinck ◽  
Matthew S. Patterson
Keyword(s):  
Abel Map

scholarly journals The Abel map for surface singularities III: Elliptic germs

2021 ◽  
Author(s):  
János Nagy ◽  
András Némethi
Keyword(s):  
General Theory ◽  
Present Note ◽  
Affine Subspace ◽  
Normal Surface ◽  
Rational Homology ◽  
Surface Singularities ◽  
Abel Map ◽  
Appl Math ◽  
Rational Homology Sphere

AbstractThe present note is part of a series of articles targeting the theory of Abel maps associated with complex normal surface singularities with rational homology sphere links (Nagy and Némethi in Math Annal 375(3):1427–1487, 2019; Nagy and Némethi in Adv Math 371:20, 2020; Nagy and Némethi in Pure Appl Math Q 16(4):1123–1146, 2020). Besides the general theory, by the study of specific families we wish to show the power of this new method. Indeed, using the general theory of Abel maps applied for elliptic singularities in this note we are able to prove several key properties for elliptic singularities (e.g. the statements of the next paragraph), which by ‘old’ techniques were not reachable. If $$({\widetilde{X}},E)\rightarrow (X,o)$$ ( X ~ , E ) → ( X , o ) is the resolution of a complex normal surface singularity and $$c_1:{\mathrm{Pic}}({\widetilde{X}})\rightarrow H^2({\widetilde{X}},{\mathbb {Z}})$$ c 1 : Pic ( X ~ ) → H 2 ( X ~ , Z ) is the Chern class map, then $${\mathrm{Pic}}^{l'}({\widetilde{X}}):= c_1^{-1}(l')$$ Pic l ′ ( X ~ ) : = c 1 - 1 ( l ′ ) has a (Brill–Noether type) stratification $$W_{l', k}:= \{{\mathcal {L}}\in {\mathrm{Pic}}^{l'}({\widetilde{X}})\,:\, h^1({\mathcal {L}})=k\}$$ W l ′ , k : = { L ∈ Pic l ′ ( X ~ ) : h 1 ( L ) = k } . In this note we determine it for elliptic singularities together with the stratification according to the cycle of fixed components. E.g., we show that the closure of any $$W(l',k)$$ W ( l ′ , k ) is an affine subspace. For elliptic singularities we also characterize the End Curve Condition and Weak End Curve Condition in terms of the Abel map, we provide several characterization of them, and finally we show that they are equivalent.

scholarly journals On the Degree-1 Abel Map for Nodal Curves

2018 ◽  
Vol 50 (3) ◽  
pp. 717-743
Author(s):  
Aldi Nestor de Souza ◽  
Frederico Sercio
Keyword(s):  
Nodal Curves ◽  
Abel Map

scholarly journals The Abel map for surface singularities I: generalities and examples

2019 ◽  
Vol 375 (3-4) ◽  
pp. 1427-1487 ◽  
Author(s):  
János Nagy ◽  
András Némethi
Keyword(s):  
Surface Singularities ◽  
Abel Map

On Algebro-Geometric Solutions of the Camassa-Holm Hierarchy

2007 ◽  
Vol 7 (3) ◽  
Author(s):  
Luca Zampogni
Keyword(s):  
Global Solutions ◽  
Natural Setting ◽  
Jacobian Variety ◽  
Generalized Jacobian ◽  
Zero Curvature ◽  
Geometric Type ◽  
Recursion Formulas ◽  
Abel Map ◽  
Sturm Liouville ◽  
Geometric Solutions

AbstractWe find global solutions of algebro geometric type for all the equations of a new commuting hierarchy containing the Camassa-Holm equation. This hierarchy is built in analogy to the classical K-dV and AKNS hierarchies. We use a zero curvature method to give recursion formulas. The time evolution of the solutions is completely determined, and the motion on a nonlinear subvariety Υ of a generalized Jacobian variety is obtained by solving an inverse problem for the Sturm-Liouville equation L(φ) = −φ″ + φ = λyφ. This is the natural setting for the expression of the solutions which depend linearly with respect to t and x, with coordinates on a curvilinear parallelogram contained in such a subvariety φ. φ is obtained as the restriction of the generalized Abel map I

PERIODIC PEAKONS AND CALOGERO–FRANÇOISE FLOWS

2005 ◽  
Vol 4 (1) ◽  
pp. 1-27 ◽  
Author(s):  
R. Beals ◽  
D. H. Sattinger ◽  
J. Szmigielski
Keyword(s):  
Spectral Problem ◽  
Shallow Water Equation ◽  
Weyl Function ◽  
Theta Functions ◽  
Hyperelliptic Curves ◽  
Completely Integrable Systems ◽  
Hamiltonian Flows ◽  
Finite Dimensional ◽  
Abel Map ◽  
Primary 35Q51

It has long been known that a number of periodic completely integrable systems are associated to hyperelliptic curves, for which the Abel map linearizes the flow (at least in part). We show that this is true for a relatively recent such system: the periodic discrete reduction of the shallow water equation derived by Camassa and Holm. The associated spectral problem has the same form and evolves in the same way as the spectral problem for a family of finite-dimensional non-periodic Hamiltonian flows introduced by Calogero and Françoise. We adapt the Weyl function method used earlier by us to solve the peakon problem to give an explicit solution to both the periodic discrete Camassa–Holm system and the (non-periodic) Calogero–Françoise system in terms of theta functions. AMS 2000 Mathematics subject classification: Primary 35Q51; 37J35; 35Q53

Quasi-periodic solutions of the Belov–Chaltikian lattice hierarchy

2017 ◽  
Vol 29 (08) ◽  
pp. 1750025 ◽  
Author(s):  
Xianguo Geng ◽  
Xin Zeng
Keyword(s):  
Asymptotic Behavior ◽  
Meromorphic Function ◽  
Periodic Solutions ◽  
Continuous Flow ◽  
Jacobian Variety ◽  
Trigonal Curve ◽  
Essential Properties ◽  
Discrete Flow ◽  
Abel Map ◽  
Matrix Spectral Problem

Utilizing the characteristic polynomial of Lax matrix for the Belov–Chaltikian (BC) lattice hierarchy associated with a [Formula: see text] discrete matrix spectral problem, we introduce a trigonal curve with three infinite points, from which we establish the associated Dubrovin-type equations. The essential properties of the Baker–Akhiezer function and the meromorphic function are discussed, that include their asymptotic behavior near three infinite points on the trigonal curve and the divisor of the meromorphic function. The Abel map is introduced to straighten out the continuous flow and the discrete flow in the Jacobian variety, from which the quasi-periodic solutions of the entire BC lattice hierarchy are obtained in terms of the Riemann theta function.

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