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Counting points with Riemann-Roch formulas

Jorge Martín-Morales

TL;DR

This work studies exact lattice-point counts in rational triangles by connecting counting functions to Riemann–Roch on weighted projective planes and to cyclic quotient singularities. It develops a residue-theoretic RR formula $L_w(d)=1+\frac{d(d+|w|)}{2\bar w}+R_w(d)$, where the correction term $R_w(d)$ is governed by local $\Delta$-invariants and Dedekind-sum-type residues, enabling precise counts via singularity theory. An efficient algorithm for computing the correction term uses a Euclidean-division recursion with worst-case Fibonacci complexity, and the noncoprime case is reduced to coprime weights. A detailed example demonstrates the full pipeline, linking intersection theory, resolution of singularities, and $\Delta$-invariants to obtain the exact count, showcasing practical impact for exact lattice-point enumeration in rational polyhedra and advancing connections between Ehrhart-type problems and singularity theory.

Abstract

We provide an algorithm for computing the number of integral points lying in certain triangles that do not have integral vertices. We use techniques from Algebraic Geometry such as the Riemann-Roch formula for weighted projective planes and resolution of singularities. We analyze the complexity of the method and show that the worst case is given by the Fibonacci sequence. At the end of the manuscript a concrete example is developed in detail where the interplay with other invariants of singularity theory is also treated.

Counting points with Riemann-Roch formulas

TL;DR

This work studies exact lattice-point counts in rational triangles by connecting counting functions to Riemann–Roch on weighted projective planes and to cyclic quotient singularities. It develops a residue-theoretic RR formula , where the correction term is governed by local -invariants and Dedekind-sum-type residues, enabling precise counts via singularity theory. An efficient algorithm for computing the correction term uses a Euclidean-division recursion with worst-case Fibonacci complexity, and the noncoprime case is reduced to coprime weights. A detailed example demonstrates the full pipeline, linking intersection theory, resolution of singularities, and -invariants to obtain the exact count, showcasing practical impact for exact lattice-point enumeration in rational polyhedra and advancing connections between Ehrhart-type problems and singularity theory.

Abstract

We provide an algorithm for computing the number of integral points lying in certain triangles that do not have integral vertices. We use techniques from Algebraic Geometry such as the Riemann-Roch formula for weighted projective planes and resolution of singularities. We analyze the complexity of the method and show that the worst case is given by the Fibonacci sequence. At the end of the manuscript a concrete example is developed in detail where the interplay with other invariants of singularity theory is also treated.
Paper Structure (21 sections, 10 theorems, 128 equations, 9 figures)

This paper contains 21 sections, 10 theorems, 128 equations, 9 figures.

Table of Contents

  1. Counting points with the residue theorem
  2. Residue at t=1
  3. Residue at t=zeta-i-l-nonzero
  4. Summary
  5. Euler characteristic of a sheaf and intersection theory in P2w
  6. The weighted projective plane
  7. Intersection theory in Pw2
  8. Euler characteristic of OPw2D
  9. The DP-invariant of a divisor
  10. Effective computation of the correction term
  11. Non-pairwise coprime case
  12. Overview with a detailed example
  13. Intersection theory and Riemann-Roch formula
  14. The correction term Rpw2D
  15. The DX-invariant
  16. ...and 6 more sections

Key Result

Theorem 1

Let $D$ be a divisor in $\mathbb{P}^2_w$ of degree $d$. Then where $\mathop{\mathrm{Sing}}\nolimits(\mathbb{P}^2_w)$ denotes the singular locus of the weighted projective plane.

Figures (9)

  • Figure 1: The five Platonic solids.
  • Figure 2: Pick's theorem for a triangle.
  • Figure 3: Vanishing of $H^i(\mathbb{P}^2_w,\mathcal{O}_{\mathbb{P}^2_w}(D))$, $i=0,1,2$.
  • Figure 4: Blowing-up at the origin of $X(d;1,q)$ with weights $(1,q)$.
  • Figure 5: Solution for $19i + 77j + 12k = 1528$. (True and rescaled picture)
  • ...and 4 more figures

Theorems & Definitions (26)

  • Theorem 1
  • Theorem 2
  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • proof
  • ...and 16 more