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Maximal Mumford Curves from Planar Graphs

Mario Kummer, Bernd Sturmfels, Raluca Vlad

TL;DR

The paper develops a novel method to construct maximal Mumford (MM) curves by deforming graph curves $C_G$ in canonical embeddings $\mathbb{P}^{g-1}$, with MM-curves existing precisely when the underlying graph $G$ is planar. It identifies a full-dimensional cone in the tangent space of the Hilbert scheme, corresponding to first-order deformations that lift to smooth Mumford curves, and shows a unique edge-pairing $\rho$ yielding $g+1$ real cycles, hence maximality. This yields full-dimensional loci of MM-curves in the moduli space $\mathcal{M}_g$ and provides explicit genus $3$–$5$ examples, together with computational tools (Macaulay2/Bertini) and an algorithm to compute the deformation directions $\eta_e$. The approach connects non-archimedean real geometry, graph-theoretic planarity via Mac Lane’s theorem, and Hilbert-scheme deformation theory to deliver a practical construction of MM-curves beyond planar plane curves. It offers a new, canonical framework for generating MM-curves and demonstrates its viability through concrete computations and explicit equations.

Abstract

A curve of genus g is maximal Mumford (MM) if it has g+1 ovals and g tropical cycles. We construct full-dimensional families of MM curves in the Hilbert scheme of canonical curves. This rests on first-order deformations of graph curves whose graph is planar.

Maximal Mumford Curves from Planar Graphs

TL;DR

The paper develops a novel method to construct maximal Mumford (MM) curves by deforming graph curves in canonical embeddings , with MM-curves existing precisely when the underlying graph is planar. It identifies a full-dimensional cone in the tangent space of the Hilbert scheme, corresponding to first-order deformations that lift to smooth Mumford curves, and shows a unique edge-pairing yielding real cycles, hence maximality. This yields full-dimensional loci of MM-curves in the moduli space and provides explicit genus – examples, together with computational tools (Macaulay2/Bertini) and an algorithm to compute the deformation directions . The approach connects non-archimedean real geometry, graph-theoretic planarity via Mac Lane’s theorem, and Hilbert-scheme deformation theory to deliver a practical construction of MM-curves beyond planar plane curves. It offers a new, canonical framework for generating MM-curves and demonstrates its viability through concrete computations and explicit equations.

Abstract

A curve of genus g is maximal Mumford (MM) if it has g+1 ovals and g tropical cycles. We construct full-dimensional families of MM curves in the Hilbert scheme of canonical curves. This rests on first-order deformations of graph curves whose graph is planar.
Paper Structure (6 sections, 15 theorems, 65 equations, 3 figures)

This paper contains 6 sections, 15 theorems, 65 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Real Algebraic Geometry with Valuations
  3. Graph Curves and Planar Graphs
  4. Hyperplane Arrangements at the Hilbert Scheme
  5. Constructing MM-Curves
  6. Computations

Key Result

Theorem 1.4

An MM-curve in $\mathbb{P}^{g-1}$ with special fiber $C_G$ exists if and only if $G$ is planar. The locus of such MM-curves is full-dimensional in the Hilbert scheme of canonical curves. Every first order deformation from an open cone, isomorphic to $\mathbb{R}^{g^2-1} \times \mathbb{R}^{3g-3}_{> 0}

Figures (3)

  • Figure 1: A genus five graph $G$ and its double-cover $G_\rho \rightarrow G$ with six cycles.
  • Figure 2: A genus five graph curve and its deformation to an MM-curve with six ovals.
  • Figure 3: The associahedron has $14$ vertices, $21$ edges and $9$ facets, labeled $p_{13},p_{14},\ldots,p_{46}$.

Theorems & Definitions (40)

  • Example 1.1: $g=3$
  • Example 1.2: $g=4$
  • Example 1.3: $g=5$
  • Theorem 1.4
  • Theorem 2.1
  • Proposition 2.2
  • proof : Sketch of Proof
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • ...and 30 more