[Paper Review] Quasi-Homogeneous Linear Systems on P^2 with Base Points of Multiplicity 6

Michael Kunte|ArXiv.org|Apr 7, 2004

Finite Group Theory Research6 references3 citations

TL;DR

This paper proves the Harbourne-Hirschowitz conjecture for quasi-homogeneous linear systems on ℙ² with base points of multiplicity 6, using degeneration techniques and results from Ciliberto-Miranda, Laface, Seibert, Ugaglia, and Yang. The key contribution is a complete classification of special systems of the form ℒ(d, m₀, 6ⁿ), showing they are precisely the (−1)-special systems, thereby solving the dimensionality problem in this case.

ABSTRACT

In this paper we prove the Harbourne-Hirschowitz conjecture for quasi-homogeneous linear systems of multiplicity 6 on P^2. For the proof we use the degeneration of the plane by Ciliberto and Miranda and results by Laface, Seibert, Ugaglia and Yang. As an application we derive a classification of the special systems of multiplicity 6.

Motivation & Objective

• To prove the Harbourne-Hirschowitz conjecture for quasi-homogeneous linear systems with base points of multiplicity 6 on ℙ².
• To classify all special linear systems ℒ(d, m₀, 6ⁿ) that fail to have the expected dimension.
• To establish that such systems are exactly the (−1)-special systems, as predicted by the conjecture.
• To use degeneration techniques and known results on (−1)-curves to verify the conjecture in this specific multiplicity case.
• To provide a complete list of all (−1)-special systems in the quasi-homogeneous multiplicity 6 setting.

Proposed method

• Employing the degeneration of the plane developed by Ciliberto and Miranda to analyze linear systems on ℙ² with base points.
• Using the virtual dimension formula v(ℒ) = d(d+3)/2 − ∑mᵢ(mᵢ+1)/2 to compute expected dimension and detect special systems.
• Applying Riemann-Roch and Serre duality on the blow-up ℙ′ of ℙ² at base points to relate ℓ(ℒ) to h¹(𝒪(𝒟)).
• Identifying (−1)-special systems via the existence of rational (−1)-curves with negative intersection numbers and non-trivial residual systems.
• Leveraging results from Laface, Seibert, Ugaglia, and Yang on special systems and Cremona transformations to verify dimensionality.
• Using computational tools like Cremona and Singular in characteristic 32003 to verify dimension and emptiness of specific systems.

Experimental results

Research questions

• RQ1 Are all special linear systems ℒ(d, m₀, 6ⁿ) on ℙ² precisely the (−1)-special systems, as predicted by the Harbourne-Hirschowitz conjecture?
• RQ2 What is the complete list of (−1)-special systems in the quasi-homogeneous case with multiplicity 6?
• RQ3 Can the degeneration method of Ciliberto and Miranda be effectively used to prove the conjecture for multiplicity 6?
• RQ4 Which systems ℒ(d, m₀, 6ⁿ) have virtual dimension ≥ 0 but projective dimension > virtual dimension, indicating speciality?
• RQ5 How do Cremona transformations and splitting off lines help in verifying the dimension of specific systems?

Key findings

• The Harbourne-Hirschowitz conjecture holds for all quasi-homogeneous linear systems ℒ(d, m₀, 6ⁿ) on ℙ².
• A complete list of (−1)-special systems is provided, with all such systems satisfying the conditions: ℒ.𝒜ᵢ = −nᵢ with nᵢ ≥ 1, some nⱼ ≥ 2, and residual system with non-negative virtual dimension.
• For ℒ(17,1,6⁸), the dimension is greater than −1, indicating it is non-special and regular, supporting the conjecture.
• Systems like ℒ(19,0,6¹⁰) and ℒ(22,0,6¹³) are empty, which helps in proving emptiness of related systems via induction.
• Computational verification using Singular in characteristic 32003 confirms dimension and emptiness for systems such as ℒ(25,12,6¹³) and ℒ(31,18,6¹⁷).
• Cremona transformations and splitting off lines are used to reduce systems like ℒ(17,7,6⁷) and ℒ(23,15,6⁹) to known regular or empty cases.