Divisor Of Degree Research Articles

Simplicial Dollar Game

The dollar game is a chip-firing game introduced by Baker as a context in which to formulate and prove the Riemann-Roch theorem for graphs. A divisor on a graph is a formal integer sum of vertices. Each determines a dollar game, the goal of which is to transform the given divisor into one that is effective (nonnegative) using chip-firing moves. We use Duval, Klivans, and Martin's theory of chip-firing on simplicial complexes to generalize the dollar game and results related to the Riemann-Roch theorem for graphs to higher dimensions. In particular, we extend the notion of the degree of a divisor on a graph to a (multi)degree of a chain on a simplicial complex and use it to establish two main results. The first of these generalizes the fact that if a divisor on a graph has large enough degree (at least as large as the genus of the graph), it is winnable; and the second generalizes the fact that trees (graphs of genus $0$) are exactly the graphs on which every divisor of degree $0$, interpreted as an instance of the dollar game, is winnable.

Pad\xe9 approximants on Riemann surfaces and KP tau functions

The paper has two relatively distinct but connected goals; the first is to define the notion of Padé approximation of Weyl–Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree g. The denominators of the resulting Padé-like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann–Hilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a “tau” function. We show how this tau function satisfies the Kadomtsev–Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2-Toda hierarchy when considering the whole sequence of tau functions. We also show how this construction is related to the Krichever construction of algebro-geometric solutions.

Moduli space of parabolic vector bundles over hyperelliptic curves

Let X be a smooth projective hyperelliptic curve of arbitrary genus g. In this article, we will classify the rank 2 stable vector bundles with parabolic structure along a reduced divisor of degree 4.

HYPERELLIPTIC GRAPHS AND METRIZED COMPLEXES

We prove a version of Clifford’s theorem for metrized complexes. Namely, a metrized complex that carries a divisor of degree $2r$ and rank $r$ (for $0<r<g-1$) also carries a divisor of degree 2 and rank 1. We provide a structure theorem for hyperelliptic metrized complexes, and use it to classify divisors of degree bounded by the genus. We discuss a tropical version of Martens’ theorem for metric graphs.

Generalized inflection points of very general effective divisors on smooth curves

Let E be a very general effective divisor of degree d on a smooth curve C of genus g. We study inflection points on linear systems |aE | for an integer a ≥ 1. They are called generalized inflection points of the invertible sheaf \({\mathcal{O}_C(E)}\). In case \({P\notin E}\) is a generalized inflection point of \({\mathcal{O}_C(E)}\) then it is a normal generalized inflection point. In case \({P\in E}\) then P has minimal vanishing sequences for E.

Cuspidal q-Rational Torsion Subgroup of J(G) of Level P

For $p\!>\!3$ an odd prime, let $\Gamma$ be a congruence subgroup between $\Gamma_1(p)$ and $\Gamma_0(p)$. In this article, we give an explicit basis for the group of modular units on $X(\Gamma)$ that have divisors defined over $\mathbb{Q}$. As an application, we determine the order of the cuspidal $\mathbb{Q}$-rational torsion subgroup of $J(\Gamma)$ generated by the divisor classes of cuspidal divisors of degree $0$ defined over $\mathbb Q$.

CHEN–RUAN COHOMOLOGY OF SOME MODULI SPACES, II

Let X be a compact connected Riemann surface of genus at least two. Let r be a prime number and ξ → X a holomorphic line bundle such that r is not a divisor of degree ξ. Let [Formula: see text] denote the moduli space of stable vector bundles over X of rank r and determinant ξ. By Γ we will denote the group of line bundles L over X such that L⊗r is trivial. This group Γ acts on [Formula: see text] by the rule (E, L) ↦ E ⊗ L. We compute the Chen–Ruan cohomology of the corresponding orbifold.

Birational rigidity and ℚ-factoriality of a singular double cover of a quadric branched over a divisor of degree 4

We prove birational rigidity and calculate the group of birational automorphisms of a nodal ℚ-factorial double cover X of a smooth three-dimensional quadric branched over a quartic section. We also prove that X is ℚ-factorial provided that it has at most 11 singularities; moreover, we give an example of a non-ℚ-factorial variety of this type with 12 simple double singularities.

Deforming Curves in Jacobians to Non-Jacobians II: Curves in C(e), 3⩽ e ⩽ g−3*

We introduce deformation theoretic methods for determining when a curve X in a nonhyperelliptic Jacobian JC will deform with JC to a non-Jacobian. We apply these methods to a particular class of curves in symmetric powers C(e) of C where 3⩽ e ⩽ g−3. More precisely, given a pencil g1d of degree d on C, let X be the curve parametrizing divisors of degree e in divisors of g1d (see the paper for the precise scheme-theoretical definition). Under certain genericity assumptions on the pair (C, g1d), we prove that if X deforms infinitesimally out of the Jacobian locus with JC then either d=2e, dim H0 (g1d)=e or d=2e+1, dim H0 (g1d)=e+1. The analogous result in the case e=2 without genericity assumptions was proved earlier.

Factorable polynomial matrices

It is shown that any regular polynomial matrix over the field of complex numbers that has at most one elementary divisor of degree 3 and whose remaining elementary divisors are of degree no greater than 2 can be factored into linear regular factors.