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Tropical curves, their Jacobians and Theta functions  [PDF]
Grigory Mikhalkin,Ilia Zharkov
Mathematics , 2006,
Abstract: 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 characteristics  [PDF]
Ilia Zharkov
Mathematics , 2007,
Abstract: This note is a follow up of math.AG/0612267v2 and it is largely inspired by a beautiful description of Baker-Norine of non-effective degree (g-1) divisors via chip-firing game. We consider the set of all theta characteristics on a tropical curve and identify the Riemann constant as a unique non-effective one among them.
Characterizing Curves by their odd Theta-characteristics  [PDF]
Lucia Caporaso,Edoardo Sernesi
Mathematics , 2002,
Abstract: We show that a general canonical curve is uniquely determined by the finite set of hyperplanes cutting theta-characteristics on it. Geometrical and combinatorial properties of the moduli space of stable spin curves are proved, which play an essential role.
Theta characteristics and their moduli  [PDF]
Gavril Farkas
Mathematics , 2012,
Abstract: We discuss various aspects of the geometry of theta characteristics including the birational geometry of the spin moduli space of curves, parametrization of moduli via special K3 surfaces, as well as the relation with classical theta function theory and string theory in the form of the superstring measure. A historical view of the development of the subject is also presented.
Theta functions of superelliptic curves  [PDF]
L. Beshaj,A. Elezi,T. Shaska
Mathematics , 2015,
Abstract: In this short survey we give a description of the theta functions of algebraic curves, half-integer theta-nulls, and the fundamental theta functions. We describe how to determine such fundamental theta functions and describe the components of the moduli space in terms of such functions.
The syntomic regulator for $K_4$ of curves  [PDF]
Amnon Besser,Rob de Jeu
Mathematics , 2012,
Abstract: Let $C$ be a curve defined over a discrete valuation field of characteristic zero where the residue field has positive characteristic. Assuming that $C$ has good reduction over the residue field, we compute the syntomic regulator on a certain part of $K_4^[(3)}(C)$. The result can be expressed in terms of $p$-adic polylogarithms and Coleman integration. We also compute the syntomic regulator on a certain part of $K_4^[(3)}(F)$ for the function field $F$ of $C$. The result can be expressed in terms of $p$-adic polylogarithms and Coleman integration, or by using a trilinear map ("triple index") on certain functions.
Orthogonal bundles, theta characteristics and the symplectic strange duality  [PDF]
Prakash Belkale
Mathematics , 2008,
Abstract: A basis for the space of generalized theta functions of level one for the spin groups, parameterized by the theta characteristics (the even theta characteristcs for the odd spin groups) on a curve, is shown to be projectively flat over the moduli space of curves (for Hitchin's connection). The symplectic strange duality conjecture, conjectured by Beauville is shown to hold for all curves of genus at least two, by using Abe's proof of the conjecture for generic curves, and the above monodromy result.
Embeddings and immersions of tropical curves  [PDF]
Dustin Cartwright,Andrew Dudzik,Madhusudan Manjunath,Yuan Yao
Mathematics , 2014,
Abstract: We construct immersions of trivalent abstract tropical curves in the Euclidean plane and embeddings of all abstract tropical curves in higher dimensional Euclidean space. Since not all curves have an embedding in the plane, we define the tropical crossing number of an abstract tropical curve to be the minimum number of self-intersections, counted with multiplicity, over all its immersions in the plane. We show that the tropical crossing number is at most quadratic in the number of edges and this bound is sharp. For curves of genus up to two, we systematically compute the crossing number. Finally, we use our immersed tropical curves to construct totally faithful nodal algebraic curves via lifting results of Mikhalkin and Shustin.
Lifting matroid divisors on tropical curves  [PDF]
Dustin Cartwright
Mathematics , 2015,
Abstract: 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 this bound might only exist over certain characteristics or over certain fields of definition. Our examples also apply to the theory of metrized complexes and weighted graphs. These examples arise by relating the lifting problem to matroid realizability. We also give a proof of Mn\"ev universality with explicit bounds on the size of the matroid, which may be of independent interest.
Prym covers, theta functions and Kobayashi curves in Hilbert modular surfaces  [PDF]
Martin Moeller
Mathematics , 2011,
Abstract: Algebraic curves in Hilbert modular surfaces that are totally geodesic for the Kobayashi metric have very interesting geometric and arithmetic properties, e.g. they are rigid. There are very few methods known to construct such algebraic geodesics that we call Kobayashi curves. We give an explicit way of constructing Kobayashi curves using determinants of derivatives of theta functions. This construction also allows to calculate the Euler characteristics of the Teichmueller curves constructed by McMullen using Prym covers.
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