arXiv subjects

Nicolas Folinsbee

Publications and source records attributed to Nicolas Folinsbee.

4 recordsLinked to original sources

arXiv2026

Duality and a Canonical Sheaf in Periodic Riemann Functions

Let $f\colon{\mathbb Z}^2\to{\mathbb Z}$ be a Riemann function whose weight $W$ is a perfect matching. Then there is a family of sheaves of $k$-vector spaces $\{{{M}}_{W,{\bf d}}\}_{{\bf d}\in{\mathbb Z}^2}$ on a five-point topological that models $f$ in that $f({\bf d})=b^0({{M}}_{W,{\bf d}})$ and that $$ b^1({{M}}_{W,{\bf d}})= f^\wedge_{\bf K}({\bf d}-{\bf K}) $$ for any ${\bf K}\in{\mathbb Z}^2$. Hence a Riemann-Roch formula for $f$ is equivalent to an Euler characteristic computation of ${{M}}_{W,{\bf d}}$. If $f$ and $W$ are $r$-periodic, then the sheaves ${{M}}_{W,{\bf d}}$ become ${{O}}_r$-modules of finite type for a natural sheaf of rings ${{O}}={{O}}_r$. We show that in this case there is a ``canonical ${{O}}$-module'' $\omega=\omega_W$ and a pairing for $i=0,1$, $$ H^i(M_{W,{\bf 0}}\otimes F) \times {\rm Ext}^{1-i}(F,M_{W^\wedge_{\bf L},{\bf K}})\to H^1(\omega)\cong k $$ that is perfect when ${\bf L}={\bf K}+{\bf 1}$ and ${{F}}$ is a certain type of line bundle or a certain type of skyscraper sheaf. In particular when ${{F}}$ is a line bundle, we realize the above formula for $b^1({{M}}_{W,{\bf d}})$ as a duality theorem akin to Serre duality. We show that canonical ${{O}}$-module $\omega_W$ is a rather exceptional element in a family of tensor products of two modules ${{M}}\otimes_{{O}}{{M}}'$, where ${{M}}$ and ${{M}}'$ vary over ${{O}}_r$-modules of the form ${{M}}_{W',{\bf d}}$. This article doesn't assume any background in sheaf theory; rather we describe all our sheaves as a ``diagrams of vector spaces,'' where each diagram is essentially a sheaf of vector spaces on a fixed topological space of five points.

math.CO↗

arXiv2022

Sheaves and Duality in the Two-Vertex Graph Riemann-Roch Theorem

For each graph on two vertices, and each divisor on the graph in the sense of Baker-Norine, we describe a sheaf of vector spaces on a finite category whose zeroth Betti number is the Baker-Norine "Graph Riemann-Roch" rank of the divisor plus one. We prove duality theorems that generalize the Baker-Norine "Graph Riemann-Roch" Theorem.

math.CO↗

arXiv2022

Euler Characteristics and Duality in Riemann Functions and the Graph Riemann-Roch Rank

By a {\em Riemann function} we mean a function $f\colon{\mathbb Z}^n\to{\mathbb Z}$ such that $f({\bf d})=f(d_1,\ldots,d_n)$ is equals $0$ for ${\rm deg}({\bf d})=d_1+\cdots+d_n$ sufficiently small, and equals $d_1+\cdots+d_n+C$ for a constant, $C$ -- the {\em offset of $f$} -- for ${\rm deg}({\bf d})$ sufficiently large. By adding $1$ to the Baker-Norine rank function of a graph, one gets an equivalent Riemann function, and similarly for related rank functions. For such an $f$, for any ${\bf K}\in{\mathbb Z}^n$ there is a unique Riemann function $f^\wedge_{\bf K}$ such that for all ${\bf d}\in{\mathbb Z}^n$ we have $$ f({\bf d}) - f^\wedge_{\bf K}({\bf K}-{\bf d}) = {\rm deg}({\bf d})+C $$ which we call a {\em generalized Riemann-Roch formula}. We show that any such equation can be viewed as an Euler charactersitic equation of sheaves of a particular simple type that we call {\em diagrams}. This article does not assume any prior knowledge of sheaf theory. To certain Riemann functions $f\colon{\mathbb Z}^2\to{\mathbb Z}$ there is a simple family of diagrams $\{\mathcal{M}_{W,{\bf d}}\}_{{\bf d}\in{\mathbb Z}^2}$ such that $f({\bf d})=b^0({\mathcal{M}}_{W,{\bf d}})$ and $f^\wedge_{\bf K}({\bf K}-{\bf d})=b^1({\mathcal{M}}_{W,{\bf d}})$. Furthermore we give a canonical isomorphism $$ H^1({\mathcal{M}}_{W,{\bf d}})^* \to H^0({\mathcal{M}}_{W',{\bf K}-{\bf d}}) $$ where $W'$ is the weight of $f^\wedge_{\bf K}$. General Riemann functions $f\colon{\mathbb Z}^2\to{\mathbb Z}$ are similarly modeled with formal differences of diagrams. Riemann functions ${\mathbb Z}^n\to{\mathbb Z}$ are modeled using their restrictions to two of their variables. These constructions involve some ad hoc choices, although the equivalence class of virtual diagram obtained is independent of the ad hoc choices.

math.CO↗

arXiv2022

Generalized Riemann Functions, Their Weights, and the Complete Graph

By a {\em Riemann function} we mean a function $f\colon{\mathbb Z}^n\to{\mathbb Z}$ such that $f({\bf d})$ is equals $0$ for $d_1+\cdots+d_n$ sufficiently small, and equals $d_1+\cdots+d_n+C$ for a constant, $C$, for $d_1+\cdots+d_n$ sufficiently large. By adding $1$ to the Baker-Norine rank function of a graph, one gets an equivalent Riemann function, and similarly for related rank functions. To each Riemann function we associate a related function $W\colon{\mathbb Z}^n\to{\mathbb Z}$ via Möbius inversion that we call the {\em weight} of the Riemann function. We give evidence that the weight seems to organize the structure of a Riemann function in a simpler way: first, a Riemann function $f$ satisfies a Riemann-Roch formula iff its weight satisfies a simpler symmetry condition. Second, we will calculate the weight of the Baker-Norine rank for certain graphs and show that the weight function is quite simple to describe; we do this for graphs on two vertices and for the complete graph. For the complete graph, we build on the work of Cori and Le Borgne who gave a linear time method to compute the Baker-Norine rank of the complete graph. The associated weight function has a simple formula and is extremely sparse (i.e., mostly zero). Our computation of the weight function leads to another linear time algorithm to compute the Baker-Norine rank, via a formula likely related to one of Cori and Le Borgne, but seemingly simpler, namely $$ r_{{\rm BN},K_n}({\bf d}) = -1+\biggl| \biggl\{ i=0,\ldots,{\rm deg}({\bf d}) \ \Bigm| \ \sum_{j=1}^{n-2} \bigl( (d_j-d_{n-1}+i) \bmod n \bigr) \le {\rm deg}({\bf d})-i \biggr\} \biggr|. $$ Our study of weight functions leads to a natural generalization of Riemann functions, with many of the same properties exhibited by Riemann functions.

math.CO↗