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Jet Schemes, Newton Polygons and Continued Fractions

Ghadi Abdallah, Maximiliano Leyton-Álvarez, Bassam Mourad, Hussein Mourtada

TL;DR

The paper develops a combinatorial/tropical framework for understanding jet schemes of Newton non-degenerate plane curve singularities. By encoding jet-component structure via staircase walks determined by the Newton polygon and Puiseux data, it proves a precise decomposition of jet schemes into hyperplane and infinite components, and shows this jet-graph encodes the embedded topological type. A canonical staircase walk $J_{SC}(f)$ recovers the jet graph from tropical data, yielding a rational generating series $G$ whose poles are explicitly tied to the Newton data. This advances connections between arc spaces, Newton polygons, and tropical geometry, providing concrete invariants and computational tools for plane curve singularities.

Abstract

We study jet schemes of Newton non-degenerate plane curve singularities. We identify a subgraph of the graph of jet components and show that it can be constructed from walks on the lattice points in the first quadrant of the Cartesian plane. In particular, we determine all the irreducible components of the jet schemes. Furthermore, we prove that this subgraph encodes the embedded topological type of the curve singularity in the plane. Finally, we introduce a generating series defined in terms of the irreducible components of the jet schemes and their (co-)dimensions, and we prove that this series is rational and explicitly determine its poles.

Jet Schemes, Newton Polygons and Continued Fractions

TL;DR

The paper develops a combinatorial/tropical framework for understanding jet schemes of Newton non-degenerate plane curve singularities. By encoding jet-component structure via staircase walks determined by the Newton polygon and Puiseux data, it proves a precise decomposition of jet schemes into hyperplane and infinite components, and shows this jet-graph encodes the embedded topological type. A canonical staircase walk recovers the jet graph from tropical data, yielding a rational generating series whose poles are explicitly tied to the Newton data. This advances connections between arc spaces, Newton polygons, and tropical geometry, providing concrete invariants and computational tools for plane curve singularities.

Abstract

We study jet schemes of Newton non-degenerate plane curve singularities. We identify a subgraph of the graph of jet components and show that it can be constructed from walks on the lattice points in the first quadrant of the Cartesian plane. In particular, we determine all the irreducible components of the jet schemes. Furthermore, we prove that this subgraph encodes the embedded topological type of the curve singularity in the plane. Finally, we introduce a generating series defined in terms of the irreducible components of the jet schemes and their (co-)dimensions, and we prove that this series is rational and explicitly determine its poles.

Paper Structure

This paper contains 8 sections, 24 theorems, 141 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Jet Schemes and Arc Spaces
  3. Special Staircase in the Plane
  4. Jet Schemes of Newton Plane Branches
  5. Jet Schemes of Newton Plane Curves
  6. Useful Results
  7. Main Theorems
  8. Generating Series

Key Result

Lemma 1

With the same notation as above, we have the following. For $k=1,\ldots,p-1,$$r_k>0$ and $r_p=0$ so that and $q=d_1+\ldots+d_p.$

Figures (4)

  • Figure 1: The walk mentioned in Question \ref{['Comb']} for $p=2$, $q=3$ and $\alpha=(0,0)$.
  • Figure 2: Different representations of the jet components graph of the cusp.
  • Figure 3: Drawing of the unweighted points representing the irreducible components of $\mathcal{C}^0_m$ : $m\in \mathbf{N}$. The red arrow corresponds to an irreducible component that continues to infinity.
  • Figure 4: Special staircase for $f=(y^2-x^3)(y^3-x^2)$

Theorems & Definitions (55)

  • Example 1
  • Example 2
  • Example 3
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Example 4
  • ...and 45 more