[Paper Review] Seshadri constants via Lelong numbers

Thomas Eckl|ArXiv.org|Aug 29, 2005

Algebraic Geometry and Number Theory14 references3 citations

TL;DR

This paper establishes a new formula for Seshadri constants on ample line bundles using Lelong numbers of singular hermitian metrics induced by sections of multiples of the line bundle. By relating the Lelong number to the minimal multiplicity of sections at a point, it derives a geometric characterization of Seshadri constants, applies it to compute constants on blowups of curve products, disproves a conjecture on maximal rationally connected quotients, and introduces a novel approach to Nagata’s conjecture via linear systems and Cremona transformations.

ABSTRACT

One of Demailly's characterizations of Seshadri constants on ample line bundles works with Lelong numbers of certain positive singular hermitian metrics. In this note sections of multiples of the line bundle are used to produce such metrics and then to deduce another formula for Seshadri constants. It is applied to compute Seshadri constants on blown up products of curves, to disprove a conjectured characterization of maximal rationally connected quotients and to introduce a new approach to Nagata's conjecture.

Motivation & Objective

• To provide a new geometric formula for Seshadri constants on ample line bundles using Lelong numbers of singular hermitian metrics.
• To compute Seshadri constants on blowups of products of curves via the new formula.
• To disprove a conjectured characterization of maximal rationally connected quotients using the new approach.
• To introduce a novel method for attacking Nagata’s conjecture based on linear systems and Cremona transformations.
• To explore the limits of the Ciliberto-Miranda recursion in improving lower bounds for Seshadri constants on the projective plane with r points.

Proposed method

• Construct singular hermitian metrics on an ample line bundle L using non-zero holomorphic sections of |kL|, with curvature current iΘh and associated Lelong number ν(Θh, x).
• Use Lemma 1.1 to relate the Lelong number ν(ϕ, x) of the metric weight to the minimal order of vanishing of sections at x: ν(ϕ, x) = min_j{ord_x σ_j}.
• Establish the equivalence ε(L, x) = sup_k { min_i{mult_x D_i} / k } over divisors D_i ∈ |kL| with x isolated in D_1 ∩ … ∩ D_n.
• Apply the formula to compute Seshadri constants on X = C_1 × … × C_r blown up at r points, using linear systems of curves with prescribed multiplicities.
• Use deformation and Cremona transformation techniques to analyze linear systems L_d(n^m) on ℙ² and prove non-emptiness and non-specialty via recursive criteria.
• Leverage the irreducibility of the parameter space of r-tuples of distinct points in ℙ² to construct divisors with isolated base points and controlled multiplicities.

Experimental results

Research questions

• RQ1 Can Seshadri constants be characterized via Lelong numbers of singular hermitian metrics induced by sections of multiples of the line bundle?
• RQ2 What are the Seshadri constants on the blowup of a product of curves at r points, and can they be computed using the new formula?
• RQ3 Does the new approach via Lelong numbers disprove the conjectured characterization of maximal rationally connected quotients?
• RQ4 Can the Ciliberto-Miranda recursion be used to improve lower bounds on Seshadri constants for r = s² + k points in ℙ²?
• RQ5 How does the new method based on Lelong numbers and linear systems contribute to Nagata’s conjecture on the Seshadri constant of the hyperplane bundle on ℙ² blown up at r points?

Key findings

• The Seshadri constant ε(L, x) equals the supremum over k and divisors D_1, ..., D_n ∈ |kL| with x isolated in their intersection, of min_i{mult_x D_i}/k.
• For the blowup of r curves at r points, the Seshadri constant satisfies ε(H; x_1, ..., x_r) ≤ 1/√r, and equality is approached when the system of curves with multiplicities m_i at x_i exists.
• The paper constructs, for any r, a divisor D ∈ |dH| with mult_{x_i} D = m_i and x_i isolated in D ∩ D' for another such divisor D', proving m_i/d_i ≤ ε(H; x_1, ..., x_r).
• Using the Ciliberto-Miranda recursion, sequences of triples (r, d_i, m_i) are constructed such that √r / a ≥ s + 1/2 for r = s² + 1, ..., s² + 4, but no improvement beyond 1/(s+1) is achieved for r = s² + k with 5 ≤ k ≤ 2s+1.
• The method provides a new framework for attacking Nagata’s conjecture by analyzing linear systems on ℙ² with prescribed multiplicities via Cremona transformations and deformation techniques.
• The irreducibility of the parameter space of r-tuples of distinct points in ℙ² ensures the existence of curves with prescribed tangent cones and multiplicities at each point, enabling the construction of isolated base points in intersections.