Tropical curves, their Jacobians and Theta functions

@article{Mikhalkin2006TropicalCT,
  title={Tropical curves, their Jacobians and Theta functions},
  author={Grigory Mikhalkin and Ilia Zharkov},
  journal={arXiv: Algebraic Geometry},
  year={2006}
}
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 and establish tropical versions of the Abel-Jacobi, Riemann-Roch and Riemann theta divisor theorems. 
Tropical theta functions and Riemann–Roch inequality for tropical Abelian surfaces
We show that the space of theta functions on tropical tori is identified with a convex polyhedron. We also show a Riemann-Roch inequality for tropical abelian surfaces by calculating theExpand
Theta characteristics on tropical curves
We give an explicit description of theta-characteristics on tropical curves and characterize the effective ones. We construct the moduli space for tropical theta-characteristics of genus g as aExpand
Moduli spaces of rational tropical curves
This note is devoted to the definition of moduli spaces of rational tropical curves with n marked points. We show that this space has a structure of a smooth tropical variety of dimension n-3. WeExpand
Lectures on Tropical Curves and Their Moduli Spaces
These are notes for a series of five lectures on “Moduli and degenerations of algebraic curves via tropical geometry” delivered at the CIMPA-CIMAT-ICTP School on Moduli of Curves, February 29–MarchExpand
Bitangents of tropical plane quartic curves
We study smooth tropical plane quartic curves and show that they satisfy certain properties analogous to (but also different from) smooth plane quartics in algebraic geometry. For example, we showExpand
Combinatorial tropical surfaces
We study the combinatorial properties of 2-dimensional tropical complexes. In particular, we prove tropical analogues of the Hodge index theorem and Noether's formula. In addition, we introduceExpand
Algebraic and tropical curves: comparing their moduli spaces
We construct the moduli space for equivalence classes of n-pointed tropical curves of genus g, together with its compactification given by weighted tropical curves, and establish some of its basicExpand
Integration over Tropical Plane Curves and Ultradiscretization
In this article we study holomorphic integrals on tropical plane curves in view of ultradiscretization. We prove that the lattice integrals over tropical curves can be obtained as a certain limit ofExpand
Tautological cycles on tropical Jacobians.
The classical Poincare formula relates the rational homology classes of tautological cycles on a Jacobian to powers of the class of Riemann theta divisor. We prove a tropical analogue of thisExpand
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
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 25 REFERENCES
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
Enumerative tropical algebraic geometry
The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon.Expand
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
Enumerative tropical algebraic geometry in R^2
The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon.Expand
Degeneration of Abelian varieties
I. Preliminaries.- II. Degeneration of Polarized Abelian Varieties.- III. Mumford's Construction.- IV. Toroidal Compactification of Ag.- V. Modular Forms and the Minimal Compactification.- VI.Expand
Affine Structures and Non-Archimedean Analytic Spaces
In this paper we propose a way to construct an analytic space over a non-archimedean field, starting with a real manifold with an affine structure which has integral monodromy. Our construction isExpand
Large Complex Structure Limits of K3 Surfaces
Motivated by the picture of mirror symmetry suggested by Strominger, Yau and Zaslow, we made a conjecture concerning the Gromov-Hausdorff limits of Calabi-Yau n-folds (with Ricci-flat K\"ahlerExpand
Complete moduli in the presence of semiabelian group action
I prove the existence, and describe the structure, of moduli space of pairs (P, Θ) consisting of a projective variety P with semiabelian group action and an ample Cartier divisor on it satisfying aExpand
Homological mirror symmetry and torus fibrations
In this paper we discuss two major conjectures in Mirror Symmetry: Strominger-Yau-Zaslow conjecture about torus fibrations, and the homological mirror conjecture (about an equivalence of the FukayaExpand
An analytic construction of degenerating abelian varieties over complete rings
© Foundation Compositio Mathematica, 1972, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditionsExpand
...
1
2
3
...