[Paper Review] Implicitization of surfaces in P^3 in the presence of base points

Laurent Busé, David Cox+1 more|ArXiv.org|May 23, 2002

Polynomial and algebraic computation19 references4 citations

TL;DR

This paper generalizes the method of moving quadrics for implicitizing rational surfaces in ℙ³ to cases with base points, showing that when the base points form a local complete intersection, the implicit equation can be computed as the resultant of the syzygies of the parametrizing ideal. The key contribution is a determinant-free, resultant-based method that extends prior work valid only in the absence of base points.

ABSTRACT

We show that the method of moving quadrics for implicitizing surfaces in P^3 applies in certain cases where base points are present. However, if the ideal defined by the parametrization is saturated, then this method rarely applies. Instead, we show that when the base points are a local complete intersection, the implicit equation can be computed as the resultant of the first syzygies.

Motivation & Objective

• To extend the method of moving quadrics for implicitizing rational surfaces in ℙ³ to cases where base points are present.
• To identify algebraic conditions under which the method remains valid despite the presence of base points.
• To address the open question from [CGZ] regarding the applicability of moving quadrics in the presence of base points.
• To provide an alternative to Gröbner bases and perturbed resultants for implicitization in geometric modeling.
• To show that when the parametrizing ideal is saturated and of degree >3, the method of moving quadrics fails, necessitating alternative approaches.

Proposed method

• Use the syzygy module of the ideal I = ⟨x,y,z,w⟩ ⊂ ℂ[s,t,u] as the foundation for constructing moving planes and quadrics.
• Construct a matrix 𝕄 whose entries are coefficients of syzygies of I and I², with determinant yielding the implicit equation under suitable conditions.
• Apply the resultant of the first syzygies of I as a method to compute the implicit equation when I is a local complete intersection.
• Leverage the regularity of homogeneous ideals to establish cohomological vanishing conditions that ensure the correctness of the resultant construction.
• Use sheaf cohomology and the exact sequence 0 → 𝒪(−1) → 𝒪 → 𝒪ℙ¹ → 0 to relate cohomology of 𝒪ℙ²(−1) to that of the ideal sheaf.
• Prove that when I is m-regular for m ≥ 2n−2 and dim(R/I)_m = deg(Z), then I is m-regular, enabling the use of syzygy-based implicitization.

Experimental results

Research questions

• RQ1 Can the method of moving quadrics be extended to rational surfaces in ℙ³ with base points?
• RQ2 Under what algebraic conditions on the parametrizing ideal I is the implicit equation still computable via moving quadrics when base points exist?
• RQ3 What happens to the method of moving quadrics when the ideal I is saturated and of degree greater than 3?
• RQ4 Can the implicit equation be recovered as a resultant of syzygies when the base points form a local complete intersection?
• RQ5 How does the number of base points affect the size and structure of the moving quadrics matrix?

Key findings

• The method of moving quadrics applies to surfaces in ℙ³ with base points if the base points form a local complete intersection and the ideal I is m-regular for m ≥ 2n−2.
• When the number of base points is at least the degree n of the parametrization, the implicit equation can be computed as the determinant of a smaller matrix than in the original [CGZ] method.
• If the parametrizing ideal I is saturated and of degree >3, the method of moving quadrics fails, as shown in Proposition 4.1.
• When I is a saturated local complete intersection, the implicit equation is equal to the resultant of a basis of the syzygy module, raised to the power of the degree n.
• The resultant of the syzygies recovers the implicit equation in the local complete intersection case, generalizing the μ-basis concept for curves.
• The condition dim(R/I)_m = deg(Z) for m ≥ 2n−2 is both necessary and sufficient for I to be m-regular, which ensures the validity of the resultant construction.