论文标题
Riemann功能和图形Riemann-Roch等级的Euler特征和二元性
Euler Characteristics and Duality in Riemann Functions and the Graph Riemann-Roch Rank
论文作者
论文摘要
通过a {\ em riemann函数}，我们的意思是一个函数$ f \ colon {\ mathbb z}^n \ to {\ mathbb z} $，这样$ f（{\ bf d}）= f（d_1，d_1，\ ldots，d_n）$等于$ 0 $ $ 0 $ $ $ bf bf d}）= d_1+\ cdots+d_n $足够小，并且等于$ d_1+\ cdots+d_n+c $，用于常数，$ c $ - {\ em off $ f $} - of $ f $} - for $ {\ rm v}（\ rm v}（{\ bf d}）$足够大。通过向图形的Baker-Norine等级函数添加$ 1 $，可以获得等效的Riemann函数，并且对于相关的等级函数类似。对于这样的$ f $，对于任何$ {\ bf k} \ in {\ mathbb z}^n $，都有一个唯一的riemann函数$ f^\ wedge _ {\ bf k} $，以便对所有$ {\ bf d} \ f^\ wedge _ {\ bf k}（{\ bf k} - {\ bf d}）= {\ rm deg}（{\ bf d}）+c $$，我们称之为{\ em广义的riemann-roch公式}。我们表明，任何此类方程都可以看作是我们称为{\ em Dibargrams}的特定简单类型的带轮的Euler字符方程。 本文不假定捆捆理论的任何先验知识。 对某些riemann功能$ f \ colon {\ mathbb z}^2 \ to {\ mathbb z} $有一个简单的图表$ \ {\ MATHCAL {M} _ {W， $ f（{\ bf d}）= b^0（{\ Mathcal {m}} _ {w，{\ bf d}}）$和$ f^\ wedge _ {\ bf k}（{\ bf k} - {\ bf k} - {\ bf d}）= b^1（{\ Mathcal {m}} _ {w，{\ bf d}}）$。此外，我们给出一个规范的同构$$ H^1（{\ Mathcal {M}} _ { $ f^\ wedge _ {\ bf k} $。 常规Riemann函数$ f \ colon {\ Mathbb z}^2 \ to {\ Mathbb Z} $类似地模拟了形式的图表差异。 riemann函数$ {\ mathbb z}^n \ to {\ mathbb z} $使用它们对两个变量的限制进行建模。这些结构涉及一些临时选择，尽管所获得的虚拟图的等效类别与临时选择无关。
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.
