[Paper Review] Inclics, galaxies, star configurations and Waldschmidt constants

Giuliana Fatabbi, Brian Harbourne+1 more|arXiv (Cornell University)|Apr 8, 2013

Commutative Algebra and Its Applications14 references3 citations

TL;DR

This paper introduces inclic schemes—unions of linear varieties with assigned multiplicities—offering an inductive method to compute Hilbert functions of their defining ideals, regardless of multiplicity choices. As a key application, it explicitly computes Waldschmidt constants for galactic inclics, extending known results on symbolic powers and star configurations in projective space.

ABSTRACT

This paper introduces complexes of linear varieties, called inclics (for INductively Constructible LInear ComplexeS). As examples, we study galaxies (these are constructed starting with a star configuration to which we add general points in a larger projective space). By assigning an order of vanishing (i.e., a multiplicity) to each member of the complex, we obtain fat linear varieties (fat points if all of the linear varieties are points). The scheme theoretic union of these fat linear varieties gives an inclic scheme $X$. For such a scheme, we show there is an inductive procedure for computing the Hilbert function of its defining ideal $I_X$, regardless of the choice of multiplicities. As an application, we show how our results allow the computation of the Hilbert functions of, for example, symbolic powers $(I_X)^{(m)}$ for arbitrary $m$ of many new examples of radical ideals $(I_X)$, and we explicitly compute the Waldschmidt constants $γ(I_X)$ for galactic inclics $X$.

Motivation & Objective

• To develop a general inductive framework for computing Hilbert functions of ideals defining unions of linear varieties with arbitrary multiplicities.
• To extend the range of computable symbolic power Hilbert functions beyond known cases, particularly for radical ideals.
• To define and study galactic inclic schemes as a special class of inclics built from star configurations and general points.
• To explicitly compute Waldschmidt constants for galactic inclics, providing asymptotic measures of initial degrees of symbolic powers.
• To generalize results on symbolic powers and Waldschmidt constants to new configurations in higher-dimensional projective spaces.

Proposed method

• Introduce inclics as inductively constructible complexes of linear subvarieties in projective space, satisfying specific containment and disjointness conditions.
• Define inclic schemes as unions of fat linear varieties, where each component is assigned a multiplicity (order of vanishing), forming a scheme-theoretic union.
• Establish a recursive procedure for computing the Hilbert function of the defining ideal $I_X$ of an inclic scheme $X$, valid for any multiplicity assignment.
• Apply the inductive Hilbert function computation to symbolic powers $(I_X)^{(m)}$ of radical ideals $I_X$, enabling computation of their Hilbert functions for arbitrary $m$.
• Define galactic inclics as inclics formed by a star configuration in a hyperplane and $h$ general points in a larger projective space, with $h=N$ for the main result.
• Use Corollary 3.5 and induction on the number of added points to compute $α((I_X)^{(j)})$, the initial degree of the $j$-th symbolic power, leading to the explicit formula for the Waldschmidt constant $\gamma(I_X)$.

Experimental results

Research questions

• RQ1 Can an inductive procedure be developed to compute the Hilbert function of the defining ideal of a scheme formed by linear varieties with arbitrary multiplicities?
• RQ2 To what extent can this inductive method be applied to compute the Hilbert functions of symbolic powers of radical ideals?
• RQ3 What is the Waldschmidt constant for a galactic inclic scheme formed by a star configuration and $N$ general points in a higher-dimensional projective space?
• RQ4 How does the initial degree of the $j$-th symbolic power of a galactic inclic ideal grow asymptotically?
• RQ5 What conditions ensure that the inductive computation of $α((I_X)^{(j)})$ stabilizes to yield a closed-form expression for the Waldschmidt constant?

Key findings

• An inductive procedure is established for computing the Hilbert function of the defining ideal of any inclic scheme, regardless of multiplicity assignments.
• The method enables explicit computation of the Hilbert functions of symbolic powers $(I_X)^{(m)}$ for arbitrary $m$ for many new examples of radical ideals $I_X$.
• For a galactic inclic $X = γ(n,N,e,u,N;S(n,e,u),\mathcal{H})$, the Waldschmidt constant is explicitly computed as $\gamma(I_X) = \frac{(N+1)u - Ne}{Nu - (N-1)e}$.
• The initial degree $\alpha((I_X)^{(j)})$ of the $j$-th symbolic power grows linearly with $j$, and the sequence $a_i = iru - (i-1)re$ satisfies $a_{i+1} = 2a_{i+1} - a_i$, with $a_i = \alpha((I_{G_i})^{(a_i)})$.
• When $e = n$, the regularity of the ideal $I_G$ equals $u - n + 1$, and $\alpha(I_G) = 2$, which helps bound the Waldschmidt constant.
• The result relies on the key observation that $\alpha(I_{a_i G_i}) = a_i + d$, where $d = a_i - a_{i-1}$, and this leads to the recurrence $a_{i+1} = a_i + (a_i - a_{i-1})$, yielding the closed-form expression for $\gamma(I_X)$.