Related papers: Rank of divisors on hyperelliptic curves and graph…

Let $\bar{G} = (G, \omega)$ be a vertex-weighted graph, and $\delta$ a divisor class on $G$. Let $r_{\bar{G}}(\delta)$ denote the combinatorial rank of $\delta$. Caporaso has introduced the algebraic rank…

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, a specialization property, and the Clifford inequality. We prove…

As in algebraic geometry, an effective divisor class on a vertex-weighted graph is called special if also its residual class is effective. We study the question, when this is true already on the level of divisors; that is, when there exists…

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's Specialization Lemma to weighted graphs.

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 the combinatorial rank is proposed, and proved in…

For a hyperelliptic curve of genus $g$, a divisor in general position of degree $g+1$ is given by polynomial equations. There is an action from an algebraic group on the representations of divisors by polynomials which fixes divisor…

We investigate, using purely combinatorial methods, structural and algorithmic properties of linear equivalence classes of divisors on tropical curves. In particular, an elementary proof of the Riemann-Roch theorem for tropical curves,…

Consider genus $g$ curves that admit degree $d$ covers to elliptic curves only branched at one point with a fixed ramification type. The locus of such covers forms a one parameter family $Y$ that naturally maps into the moduli space of…

For an (imaginary) hyperelliptic curve $\mathcal{H}$ of genus $g$, we determine a basis of the Riemann-Roch space $\mathcal{L}(D)$, where $D$ is a divisor with positive degree $n$, linearly equivalent to $P_1+\cdots+ P_j+(n-j)\Omega$, with…

We study the problem of computing the rank of a divisor on a finite graph, a quantity that arises in the Riemann-Roch theory on a finite graph developed by Baker and Norine (Advances of Mathematics, 215(2): 766-788, 2007). Our work consists…

Let X be a smooth complex projective curve of genus g bigger or equal to 1. If g is bigger than 1 assume further that X is either bielliptic or with general moduli. Under a natural condition on slopes, we prove that there exists a short…

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…

Let E and F be vector bundles over a complex projective smooth curve X, and suppose that 0 -> E -> W -> F -> 0 is a nontrivial extension. Let G be a subbundle of F, and D an effective divisor on X. We give a criterion for the subsheaf G(-D)…

In the classical case of irreducible smooth algebraic curves every genus $2$ curve is hyperelliptic, or in other words there is a complete linear series $g_2^1$ on them. On the other hand if $g > 2$, then a generic smooth curve of genus $2$…

Let $\mathcal{X} : y^2 = f(x)$ be a hyperelliptic curve over $\mathbb{Q}(T)$ of genus $g\geq 1$. Assume that the jacobian of $\mathcal{X}$ over $\mathbb{Q}(T)$ has no subvariety defined over $\mathbb{Q}$. Denote by $\mathcal{X}_t$ the…

In an influential 2008 paper, Baker proposed a number of conjectures relating the divisor theory of algebraic curves with an analogous combinatorial theory on finite graphs. In this note, we examine Baker's Brill--Noether existence…

If G is a finite subgroup of the automorphism group of a projective curve X and D is a divisor on X stabilized by G, then under the assumption that D is nonspecial, we compute a simplified formula for the trace of the natural representation…

Baker and Norine initiated the study of graph divisors as a graph-theoretic analogue of the Riemann-Roch theory for Riemann surfaces. One of the key concepts of graph divisor theory is the {\it rank} of a divisor on a graph. The importance…

Let R be any subring of the reals. We present a generalization of linear systems on graphs where divisors are R-valued functions on the set of vertices and graph edges are permitted to have nonegative weights in R. Using this…

In this paper we study the $X$-rank of points with respect to smooth linearly normal curves $X\subset \PP n$ of genus $g$ and degree $n+g$. We prove that, for such a curve $X$, under certain circumstances, the $X$-rank of a general point of…