[Paper Review] Local effectivity in projective spaces

Marcin Dumnicki, Tomasz Szemberg+1 more|arXiv (Cornell University)|Feb 23, 2018

Algebraic Geometry and Number Theory11 references3 citations

TL;DR

This paper introduces the Waldschmidt decomposition, a novel method to recursively bound Waldschmidt constants of very general points in projective spaces by decomposing divisors based on effectivity rather than nefness. It establishes effective lower bounds that verify Demailly's conjecture in new cases, including for $̂{\alpha}(\mathbb{P}^4;180) \geq 3.495$ and improved bounds for $\mathbb{P}^5$ with as few as 1649 points.

ABSTRACT

In this note we introduce a Waldschmidt decomposition of divisors which might be viewed as a generalization of Zariski decomposition based on the effectivity rather than the nefness of divisors. As an immediate application we prove a recursive formula providing new effective lower bounds on Waldschmidt constants of very general points in projective spaces. We use these bounds in order to verify Demailly's conjecture in a number of new cases.

Motivation & Objective

• To develop a new recursive method for bounding Waldschmidt constants of very general points in projective spaces.
• To generalize Zariski decomposition by focusing on effectivity rather than nefness, introducing the Waldschmidt decomposition of divisors.
• To provide effective lower bounds on Waldschmidt constants that verify Demailly's conjecture in previously unverified cases.
• To offer computationally efficient algorithms for estimating Waldschmidt constants using heuristic distribution strategies.
• To establish general effective lower bounds via combinatorial optimization of point distributions across hyperplanes.

Proposed method

• Introduce the Waldschmidt decomposition of divisors as a generalization of Zariski decomposition, based on the effectivity of $f^*L - tE$ rather than its nefness.
• Define the $\mu$-invariant and its reciprocal, the Waldschmidt constant $\widehat{\alpha}(X;L,Z)$, as a measure of local effectivity.
• Use recursive descent: decompose a divisor on $\mathbb{P}^N$ into $k+1$ divisors on $\mathbb{P}^{N-1}$, optimizing the distribution of multiplicities.
• Apply a heuristic strategy to minimize $\sum_{j=1}^k \frac{1}{\sqrt[N-1]{r_j}}$ under the constraint $k r_1 + r_{k+1} \leq r$, favoring nearly equal $r_j$ values.
• Implement a two-tiered algorithm: one for fast computation (bound) and one for tighter bounds (boundmore), using Singular.
• Derive general bounds via combinatorial optimization, e.g., distributing $r$ points across $k+1$ hyperplanes with $s$ groups of size $(k+1)^{N-1}$ and $k-s$ of size $k^{N-1}$.

Experimental results

Research questions

• RQ1 Can a recursive decomposition method based on effectivity improve lower bounds on Waldschmidt constants in projective spaces?
• RQ2 To what extent can the Waldschmidt decomposition generalize Zariski decomposition by replacing nefness with effectivity?
• RQ3 Can effective lower bounds on Waldschmidt constants be derived that verify Demailly’s conjecture in new cases?
• RQ4 How can computational efficiency be improved while maintaining tight bounds in high-dimensional projective spaces?
• RQ5 What optimal distribution of points across hyperplanes yields the strongest lower bounds on $\widehat{\alpha}(\mathbb{P}^N;r)$?

Key findings

• The Waldschmidt decomposition provides a recursive method to bound $\widehat{\alpha}(\mathbb{P}^N;r)$ using descent to $\mathbb{P}^{N-1}$ with optimized multiplicity distributions.
• For $\mathbb{P}^4$ with $r = 180$, the bound $\widehat{\alpha}(\mathbb{P}^4;180) \geq 3.495$ is achieved, specifically $430502824/123159135$.
• Using Proposition 4.4, $\widehat{\alpha}(\mathbb{P}^5;1649) \geq 4 + \frac{1}{5} = 4.2$ is established, requiring only 1649 points instead of 3125 for the next integer bound.
• For $\mathbb{P}^5$, $\widehat{\alpha}(\mathbb{P}^5;2018) \geq 4.4$, $\widehat{\alpha}(\mathbb{P}^5;2387) \geq 4.6$, and $\widehat{\alpha}(\mathbb{P}^5;2756) \geq 4.8$ are verified with minimal point counts.
• The heuristic method of equalizing $r_j$ values minimizes $\sum \frac{1}{\sqrt[N-1]{r_j}}$, leading to tighter bounds with reduced computational cost.
• The algorithmic implementation in Singular (bound and boundmore) enables scalable computation of bounds even for $N \geq 100$, with boundmore yielding superior results at the cost of longer runtime.