[Paper Review] Quasi-homogeneous linear systems on P2 with base points of multiplicity 7, 8, 9, 10

Marcin Dumnicki|ArXiv.org|Apr 8, 2008

Algebraic Geometry and Number Theory1 references4 citations

TL;DR

This paper proves the Harbourne-Hirschowitz conjecture for quasi-homogeneous linear systems on the projective plane with base points of multiplicity 7, 8, 9, and 10. Using a reduction algorithm, Cremona transformations, point gluing techniques, and direct matrix rank computations over finite fields, the authors establish non-specialty and non-emptiness for these systems, extending previous results for lower multiplicities and providing a more efficient method than degeneration techniques.

ABSTRACT

In the paper we prove Harbourne-Hirschowitz conjecture for quasi-homogeneous linear systems on $\mathbb P^2$ for $m=7$, 8, 9, 10, i.e. systems of curves of given degree passing through points in general position with multiplicities at least $m,...,m,m_0$, where $m=7$, 8, 9, 10, $m_0$ is arbitrary.

Motivation & Objective

• To prove the Harbourne-Hirschowitz conjecture for quasi-homogeneous linear systems with multiplicities 7, 8, 9, and 10 on ℙ².
• To extend the known validity of the conjecture beyond previous bounds (m ≤ 6) for quasi-homogeneous systems.
• To provide a systematic method for verifying non-specialty and non-emptiness of such systems without relying on degeneration techniques.
• To demonstrate the effectiveness of the reduction algorithm and diagram cutting method for handling large systems with high multiplicities.

Proposed method

• The reduction algorithm from [Dum 07b] and [Dum–Jar 07] is used to transform systems into simpler forms, enabling the detection of emptiness or non-specialty.
• Cremona transformations are applied to simplify systems and preserve dimension and speciality properties.
• The 'glueing' of points is used to reduce systems with many base points to smaller, more manageable configurations.
• Direct computation of matrix rank over 𝔽ₚ is used to verify non-emptiness for systems where theoretical reduction fails, with maximal rank implying non-emptiness over ℂ.
• The diagram cutting method from [Dum 07a] is applied to verify non-specialty for specific systems by visual decomposition.
• A combination of theoretical reductions and computational verification is used to cover all remaining cases not resolved by algebraic methods.

Experimental results

Research questions

• RQ1 Does the Harbourne-Hirschowitz conjecture hold for quasi-homogeneous linear systems on ℙ² with base point multiplicities 7, 8, 9, and 10?
• RQ2 Can the reduction algorithm replace degeneration techniques in proving non-specialty for high-multiplicity systems?
• RQ3 Are Cremona transformations and point-gluing techniques effective in simplifying systems with multiplicities up to 10?
• RQ4 Can matrix rank computations over finite fields reliably determine non-emptiness of large linear systems in characteristic zero?
• RQ5 Is the diagram cutting method sufficient to verify non-specialty without full matrix computation?

Key findings

• The Harbourne-Hirschowitz conjecture holds for all quasi-homogeneous linear systems on ℙ² with multiplicities 7, 8, 9, and 10, and arbitrary additional multiplicity m₀.
• The system ℒ(35;16,10×9) is non-empty and non-special, verified via a sequence of 12- and 10-reductions followed by diagram enlargement and reduction to empty.
• The system ℒ(28;12,8×10) is non-special, confirmed by diagram cutting and matrix rank computation over 𝔽ₚ.
• The system ℒ(13;5,4×9) is non-special, and its non-specialty is re-proven via diagram cutting, eliminating the need for large matrix computation.
• The reduction algorithm enables efficient verification of non-specialty across a broad class of systems, significantly reducing reliance on computational matrix rank evaluation.
• The method successfully handles four higher multiplicities (7–10) in a single framework, outperforming prior degeneration-based approaches in efficiency and scope.