Specialization of linear systems from curves to graphs

@article{Baker2007SpecializationOL,
  title={Specialization of linear systems from curves to graphs},
  author={Matthew Baker},
  journal={Algebra \& Number Theory},
  year={2007},
  volume={2},
  pages={613-653}
}
  • M. Baker
  • Published 2 January 2007
  • Mathematics
  • Algebra & Number Theory
We investigate the interplay between linear systems on curves and graphs in the context of specialization of divisors on an arithmetic surface. We also provide some applications of our results to graph theory, arithmetic geometry, and tropical geometry. 

Figures from this paper

Rank of divisors on graphs: an algebro-geometric analysis
The divisor theory for graphs is compared to the theory of linear series on curves through the correspondence associating a curve to its dual graph. An algebro-geometric interpretation of theExpand
LINEAR SYSTEMS ON GRAPHS WITH A REAL STRUCTURE
A degeneration of a smooth projective curve to a strongly stable curve gives rise to a specialization map from divisors on curves to divisors on graphs. In this paper we show that this specializationExpand
Linear pencils on graphs and on real curves
A degeneration of curves gives rise to an interesting relation between linear systems on curves and on graphs. In this paper, we consider the case of linear pencils and as an application, we obtainExpand
Riemann–Roch theory for weighted graphs and tropical curves
Abstract We define a divisor theory for graphs and tropical curves endowed with a weight function on the vertices; we prove that the Riemann–Roch theorem holds in both cases. We extend Baker’sExpand
Algebraic and combinatorial rank of divisors on finite graphs
Abstract We study the algebraic rank of a divisor on a graph, an invariant defined using divisors on algebraic curves dual to the graph. We prove it satisfies the Riemann–Roch formula, aExpand
Rank of divisors on tropical curves
TLDR
This work confirms a conjecture of Baker asserting that the rank of a divisor D on a (non-metric) graph is equal to theRank of D on the corresponding metric graph, and constructs an algorithm for computing therank of a Divisor on a tropical curve. Expand
Degeneration of Linear Series from the Tropical Point of View and Applications
We discuss linear series on tropical curves and their relation to classical algebraic geometry, describe the main techniques of the subject, and survey some of the recent major developments in theExpand
A specialization inequality for tropical complexes
We prove a specialization inequality relating the dimension of the complete linear series on a variety to the tropical complex of a regular semistable degeneration. Our result extends Baker'sExpand
Lifting matroid divisors on tropical curves
Tropical geometry gives a bound on the ranks of divisors on curves in terms of the combinatorics of the dual graph of a degeneration. We show that for a family of examples, curves realizing thisExpand
A Note on Jacobians, Tutte Polynomials, and Two-Variable Zeta Functions of Graphs
We address questions posed by Lorenzini about relations between Jacobians, Tutte polynomials, and the Brill–Noether theory of finite graphs, as encoded in his two-variable zeta functions. InExpand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 107 REFERENCES
Graphs, arithmetic surfaces, and the Riemann-Roch theorem
We use the theory of arithmetic surfaces to show that the Riemann-Roch theorem for Q-graphs is a direct consequence of the usual Riemann-Roch theorem for curves in algebraic geometry.
Algebraic Geometry and Arithmetic Curves
Introduction 1. Some topics in commutative algebra 2. General Properties of schemes 3. Morphisms and base change 4. Some local properties 5. Coherent sheaves and Cech cohmology 6. Sheaves ofExpand
Tropical curves, their Jacobians and Theta functions
We study Jacobian varieties for tropical curves. These are real tori equipped with integral affine structure and symmetric bilinear form. We define tropical counterpart of the theta function andExpand
Tropical geometry and its applications
From a formal perspective tropical geometry can be viewed as a branch of geometry manipulating with certain piecewise-linear objects that take over the role of classical algebraic varieties. ThisExpand
A Riemann–Roch theorem in tropical geometry
Recently, Baker and Norine have proven a Riemann–Roch theorem for finite graphs. We extend their results to metric graphs and thus establish a Riemann–Roch theorem for divisors on (abstract) tropicalExpand
Local Rings
In this chapter we will study the local rings O P ; in particular we will see that the algebraic structure of O P contains information about whether a point P is singular or not, and we will learnExpand
Algebraic Geometry
Introduction to Algebraic Geometry.By Serge Lang. Pp. xi + 260. (Addison–Wesley: Reading, Massachusetts, 1972.)
On the indices of curves over local fields
Fix a non-negative integer g and a positive integer I dividing 2g − 2. For any Henselian, discretely valued field K whose residue field is perfect and admits a degree I cyclic extension, we constructExpand
Periods of Hilbert modular forms and rational points on elliptic curves
Let E be a modular elliptic curve over a totally real field. Chapter 8 of [Dar2] formulates a conjecture allowing the construction of canonical algebraic points on E by suitably integrating theExpand
Rational Points on Modular Elliptic Curves
Elliptic curves Modular forms Heegner points on $X_0(N)$ Heegner points on Shimura curves Rigid analytic modular forms Rigid analytic modular parametrisations Totally real fields ATR pointsExpand
...
1
2
3
4
5
...