[Paper Review] An effective criterion for algebraic contractibility of rational curves

Pinaki Mondal|arXiv (Cornell University)|Jan 1, 2013

Algebraic Geometry and Number Theory28 references5 citations

TL;DR

This paper presents an effective, non-numerical criterion for algebraic contractibility of rational curves in the context of rational surfaces, specifically for the union of a strict transform of a line and all but one exceptional divisor in a blow-up of ℂℙ². The criterion, based on global variants of MacLane's key polynomials, is both necessary and sufficient, and it establishes a correspondence between normal algebraic compactifications of ℂ² with one curve at infinity and algebraic curves in ℂ² with one place at infinity.

ABSTRACT

Let f: Y -> CP^2 be a birational morphism of non-singular (rational) surfaces. We give an effective (necessary and sufficient) criterion for algebraicity of the surfaces resulting from contraction of the union of the strict transform of a line on CP^2 and all but one of the exceptional divisors of f. As a by-product we construct normal non-algebraic Moishezon surfaces with the `simplest possible' singularities, which in particular completes the answer to a remark of Grauert. Our criterion involves `global variants' of `key polynomials' introduced by MacLane. The geometric formulation of the criterion yields a correspondence between normal algebraic compactifications of C^2 with one irreducible curve at infinity and algebraic curves in C^2 with one place at infinity.

Motivation & Objective

• To address the gap in effective criteria for algebraic contractibility of rational curves, particularly in cases where existing numerical criteria fail.
• To provide a necessary and sufficient condition for algebraic contractibility in a specific geometric setup: contraction of the strict transform of a line and all but one exceptional divisor in a blow-up of ℂℙ².
• To construct new examples of non-algebraic normal Moishezon surfaces with 'simplest possible' singularities, completing a question posed by Grauert.
• To demonstrate that key polynomials—originally local tools—retain global geometric information when computed in global coordinates.
• To establish a new geometric correspondence between normal algebraic compactifications of ℂ² with one curve at infinity and algebraic curves in ℂ² with one place at infinity.

Proposed method

• Introduces global variants of MacLane's key polynomials, adapted to the global geometry of the surface and the curve to be contracted.
• Uses a weighted grading on the coordinate ring of ℂ² to define a filtration and associated initial forms, enabling the construction of key polynomials in a global setting.
• Applies Buchberger's algorithm to prove that a certain set of polynomials forms a Gröbner basis with respect to a monomial order, ensuring effective computation.
• Establishes a surjective homomorphism from the coordinate ring of the surface to a graded ring, with kernel identified via the ideal generated by key polynomials.
• Employs the positivity of the weight function and properties of key forms to ensure that the minimal degree elements in the kernel are uniquely determined.
• Leverages the structure of the exceptional divisor and the strict transform of the line to define a birational morphism and analyze the contraction process.

Experimental results

Research questions

• RQ1 When is the union of the strict transform of a line and all but one exceptional divisor in a blow-up of ℂℙ² algebraically contractible?
• RQ2 Can an effective (computable) criterion for algebraic contractibility be given in cases where numerical invariants are insufficient?
• RQ3 What is the geometric relationship between normal algebraic compactifications of ℂ² with one curve at infinity and algebraic curves in ℂ² with one place at infinity?
• RQ4 Can non-algebraic normal Moishezon surfaces be constructed by contracting trees of rational curves, and what are the simplest possible singularities they may possess?
• RQ5 How do MacLane's local key polynomials retain global geometric information when used in a global coordinate system?

Key findings

• The paper provides the first effective and necessary-and-sufficient criterion for algebraic contractibility in the specified geometric setting, resolving a long-standing gap in the literature.
• The criterion is based on the construction of global key polynomials and their associated Gröbner basis structure, which ensures computability and effectiveness.
• The criterion establishes a one-to-one correspondence between normal algebraic compactifications of ℂ² with one irreducible curve at infinity and algebraic curves in ℂ² with one place at infinity.
• The authors construct new examples of non-algebraic normal Moishezon surfaces with singularities that are minimal in complexity, answering a question posed by Grauert.
• The method demonstrates that key polynomials, originally defined for local valuation theory, carry global geometric information when used in a global coordinate system.
• The proof shows that the kernel of a certain ring homomorphism is generated by the ideal of key polynomials, and that this ideal is precisely the defining ideal of the contracted surface.