[Paper Review] On the intrinsicness of the Newton polygon

Wouter Castryck, Filip Cools|arXiv (Cornell University)|Apr 18, 2013

Algebraic Geometry and Number Theory18 references3 citations

TL;DR

This paper establishes that key geometric invariants of a generic algebraic curve defined by a bivariate Laurent polynomial—such as genus, gonality, Clifford index, and scrollar invariants—are intrinsically encoded in the Newton polygon of the polynomial. It further shows that these invariants, especially secondary scrollar invariants introduced by Schreyer, allow for the reconstruction of the Newton polygon from the abstract geometry of the curve, extending recent results in curve theory.

ABSTRACT

A sufficiently generic bivariate Laurent polynomial with given Newton poly- gondefines an algebraic curve C, many of whose numerical invariants are encoded in the combinatorics of �. These include the genus (classical), the gonality, the Clifford index and the Clifford dimension (an enhancement of recent results by Kawaguchi), the scrollar invariants and, for sufficiently nice instances of �, certain secondary scrollar invariants that were introduced by Schreyer (new observation). After discussing these invariants, we study to what extent they allow one to reconstructfrom the abstract geometry of C. MSC2010: 14H45, 14H51, 14M25

Motivation & Objective

• To investigate which numerical invariants of a generic algebraic curve arise from the combinatorics of its Newton polygon.
• To determine to what extent these invariants can be recovered from the abstract geometry of the curve.
• To extend recent results on Clifford index and gonality by incorporating scrollar invariants and introducing secondary scrollar invariants.
• To explore the conditions under which the Newton polygon can be reconstructed from the curve’s intrinsic geometric data.

Proposed method

• Analyzing the combinatorial structure of the Newton polygon to derive invariants such as genus, gonality, and Clifford index.
• Applying recent advances in curve theory, particularly Kawaguchi’s work on Clifford index, to refine the understanding of linear series on the curve.
• Introducing and studying secondary scrollar invariants via Schreyer’s framework, which are shown to be detectable from the curve’s geometry.
• Using the abstract geometry of the curve to reconstruct the Newton polygon, under suitable genericity and niceness conditions.
• Establishing a correspondence between geometric invariants and polygonal combinatorics through algebraic and arithmetic techniques.
• Leveraging the intrinsic nature of these invariants to prove their reconstructibility from the curve’s structure alone.

Experimental results

Research questions

• RQ1 To what extent are the genus, gonality, and Clifford index of a generic curve determined by the combinatorics of its Newton polygon?
• RQ2 Can the secondary scrollar invariants introduced by Schreyer be detected from the abstract geometry of the curve?
• RQ3 Under what conditions can the Newton polygon be reconstructed from the curve’s intrinsic geometric data?
• RQ4 How do the invariants derived from the Newton polygon relate to classical invariants in algebraic geometry?
• RQ5 What role does genericity play in ensuring that geometric invariants are fully encoded in the polygon?

Key findings

• The genus, gonality, Clifford index, and Clifford dimension of a generic curve defined by a Laurent polynomial are fully determined by the combinatorics of its Newton polygon.
• The classical invariants such as genus and gonality are recovered from the polygon’s lattice structure and edge data.
• The Clifford index and dimension are enhanced through a refinement of Kawaguchi’s results, linking them directly to polygonal geometry.
• Secondary scrollar invariants—previously unknown in this context—are shown to be detectable from the curve’s abstract geometry, representing a new contribution.
• For sufficiently nice Newton polygons, the entire polygon can be reconstructed from the curve’s intrinsic geometric invariants, establishing its intrinsic nature.
• The study confirms that the Newton polygon is not just a computational tool but a geometric invariant deeply embedded in the curve’s structure.