The Assymptotic Invariants of a Fermat-Type Set of Points in ${\mathbb{P}}^{\mathbf{3}}$

Abstract

In this paper, we compute asymptotic invariants—specifically, the Waldschmidt constants and the Seshadri constants—of a set of 31 points in ${\mathbb{P}}^3$, defined as the intersection points of a Fermat-type arrangement of planes.

1. Introduction

In recent years, there has been increasing activity in computing various invariants of specific, typically highly symmetric configurations of points in projective spaces. For example, in [1], Pokora computed the Seshadri constants of singular points of Severi curves (irreducible plane curves with the maximal number of nodes), star configurations (see [2] for definitions), and the singular points of the Klein and Wiman line arrangements (see [3] for a detailed discussion of these line arrangements and additional references). The study was extended in [4], particularly to the singular locus of the Hesse arrangement.

In [1], Pokora raised the question of whether the multi-point Seshadri constants of singular points in planar line arrangements are computed by the lines in the arrangement. This question was reiterated in a recent preprint by Hanumanthu, Kumar Roy, and Subramanian [5], where the authors extend the investigation from line arrangements to what they call transversal arrangements.

The results mentioned so far concern only Seshadri constants, which capture the local positivity of line bundles. While Seshadri constants originated in the study of vector bundles and were later transplanted into the context of linear series by Demailly [6], there exists another invariant, the Waldschmidt constant, which effectively captures the local effectivity of line bundles. Interestingly, this invariant has its origins in complex analysis and the works of Waldschmidt [7]. The interest in Waldschmidt constants stems partly from the Chudnovsky Conjecture (stated in Section 3) and partly from its interaction with other fundamental invariants such as the Castelnuovo–Mumford regularity and resurgence numbers.

Waldschmidt constants have been studied from various perspectives. In [8], Farnik and co-authors examined planar point sets with small Waldschmidt constants and established strong geometric constraints on such sets. In [9], Malara, Szpond, and the second author provided lower bounds on the Waldschmidt constants of very general point sets in ${\mathbb{P}}^N$. These bounds were significantly improved in the recent work of Dumnicki, Szpond, and the second author [10]. Independently, the Waldschmidt constants of general or very general points in projective spaces have been investigated in [11,12,13], among others. The Waldschmidt constants of the previously mentioned Klein and Wiman configurations were studied by Bauer and co-authors in [3]. In a more recent study [14], Calvo investigated the Waldschmidt constant of a unique configuration of 31 points in the projective plane, which arose as singularities of an arrangement of 15 lines.

This line of research has not been extensively pursued in higher-dimensional projective spaces, primarily due to the lack of general methods and significantly increased difficulties. This gap has motivated our investigation in the present paper.

The choice of the 31 points was not completely arbitrary; they were prominently featured in [15], where the authors constructed the first example of an unexpected (in the sense of [16]) subvariety of a dimension greater than 1.

Turning to details, our main results are Theorems 1 and 2, which provide $\widehat{\alpha }\left(Z\right)=3$ for the Waldschmidt constant and $\epsilon ({\mathbb{P}}^3;{\mathcal{O}}_{{\mathbb{P}}^3}\left(1\right),Z)={\displaystyle \frac{1}{4}}$ for the Seshadri constant.

Throughout this paper, we work over the field $\mathbb{C}$ of complex numbers.

2. Fermat-Type Arrangements and Configurations

Fermat arrangements of lines in ${\mathbb{P}}^2$ are a special case of reflection arrangements, i.e., arrangements defined by symmetry axes of reflections generating a reflection group (see [17] for details and the classification). They were studied under the name of Ceva arrangements by Hirzebruch [18]. The term Fermat arrangements was introduced by Urzúa in his thesis (see also [19]). The definition was extended to arrangements of planes in ${\mathbb{P}}^3$ by Malara and Szpond in [20] and to arrangements of hyperplanes in higher-dimensional projective space by Szpond [21].

Let us recall some definitions.

Definition 1 (The Fermat-type arrangement of hyperplanes).

The Fermat-type arrangement ${\mathcal{F}}_N^n$ is the set of hyperplanes in ${\mathbb{P}}^N$ defined by the linear factors of the polynomial

$$F_{N,n}(x_0:\dots :x_N)=\prod _{0⩽i<j⩽N}(x_i^n-x_j^n).$$

To such an arrangement, there is a naturally associated configuration of its singular points.

Definition 2 (The Fermat-type configuration of points).

The Fermat-type configuration of points is the finite set of the points of the locally (in the complex topology) maximal multiplicity in the arrangement ${\mathcal{F}}_N^n$.

In this paper, we focus on the Fermat-type configuration of points determined by the arrangement ${\mathcal{F}}_3^3$. To simplify the notation, it is convenient to work with the projective coordinates $(x:y:z:w)$ rather than $(x_0:x_1:x_2:x_3)$. In these coordinates we have

$$F_{3,3}=(x^3-y^3)(x^3-z^3)(x^3-w^3)(y^3-z^3)(y^3-w^3)(z^3-w^3)$$

(1)

and the associated configuration of points, Z, is the union of a set of 27 points, X, forming a complete intersection of the type $(3,3,3)$ and the four coordinate points, Y.

The points in X have the following coordinates:

$$(1:{\epsilon }^{\alpha }:{\epsilon }^{\beta }:{\epsilon }^{\gamma }),$$

(2)

where $\epsilon$ is a primitive root of unity of the order 3 and $0⩽\alpha ,\beta ,\gamma ⩽2$. The ideal of X is

$$I\left(X\right)=\langle x^3-y^3,y^3-z^3,z^3-w^3\rangle .$$

The ideal of the coordinate points is

$$I\left(Y\right)=\langle xy,xz,xw,yz,yw,zw\rangle .$$

Intersecting these ideals, we obtain the ideal of Z; see [15] (Lemma 1) for details.

Lemma 1 (Ideal of Z).

The ideal $I\left(Z\right)$ of the 31 points is generated in a single degree, 4, by the forms

$$\begin{array}{cccc}x(y^3-z^3),& x(z^3-w^3),& y(x^3-z^3),& y(z^3-w^3),\\ z(x^3-y^3),& z(y^3-w^3),& w(x^3-y^3),& w(y^3-z^3).\end{array}$$

3. Local Effectivity

Let U be a (non-necessarily irreducible) subvariety of a smooth projective variety, V, and let $f:{\mathrm{Bl}}_{U}V\to V$ be the blow-up along U with the exceptional divisor E. Let L be an ample line bundle on V. There is a natural numerical invariant associated with the triple $(V;L,U)$, defined as follows:

Definition 3 (The μ-invariant).

The real number

$$\mu (V;L,U)=sup\left\{t\in \mathbb{R}:f^{*}L-tEis effective\right\}$$

is the $\mu$-invariant of L along U.

This invariant measures how far $f^{*}L$ is from the boundary of the effective cone on ${\mathrm{Bl}}_{U}V$ in the direction of the exceptional divisor E.

While the $\mu$-invariant $\mu (V;L,U)$ is not commonly found in the literature, its reciprocal corresponds to the well-known Waldschmidt constant of U. To begin, recall that the initial degree of U with respect to L is defined as

$$\alpha (V;L,U)=min\left\{d:d{f}^{*}L-E\mathrm{i}\mathrm{s} \mathrm{e}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\right\}.$$

For an integer $m⩾1$, let $mU$ denote the subscheme defined by the symbolic power $I{\left(U\right)}^{\left(m\right)}$ of the saturated ideal $I\left(U\right)$ of U; see [22] (Definition 9.3.4). The asymptotic version of the initial degree is given by the following definition:

Definition 4 (Waldschmidt constant).

Let V be a smooth projective variety and let L be an ample line bundle on V. Let $U\subset V$ be a subscheme. The real number

$$\widehat{\alpha }(V;L,U)=\underset{m⩾1}{inf}{\displaystyle \frac{\alpha (V;L,mU)}{m}}$$

is called the Waldschmidt constant of U with respect to L.

Remark 1.

It is straightforward to see that the numbers $\alpha (V;L,mU)$ for $m⩾1$ form a subadditive sequence, i.e.,

$$\alpha (V;L,(k+\ell \left)U\right)⩽\alpha (V;L,kU)+\alpha (V;L,\ell U)$$

for all k and ℓ. Thus, the infimum in Definition 4 exists, and moreover, we have

$$\widehat{\alpha }(V;L,U)=\underset{m\to \infty }{lim}{\displaystyle \frac{\alpha (V;L,mU)}{m}}.$$

In this paper, we focus on a finite set of points, U, in projective space. Since we always take $L={\mathcal{O}}_{{\mathbb{P}}^N}\left(1\right)$, we omit $V={\mathbb{P}}^N$ and L from the notation. The initial degree of the ideal $I\left(U\right)$ of U is then defined as

$$\alpha \left(I\left(U\right)\right)=min\left\{d:{\left[I\left(U\right)\right]}_d\ne 0\right\},$$

where ${\left[I\left(U\right)\right]}_d$ denotes the degree (d) component of $I\left(U\right)$.

Accordingly, the Waldschmidt constant of an ideal $I\subset \mathbb{C}[x_0,\dots ,x_N]$ then satisfies

$$\widehat{\alpha }\left(I\right)=\underset{m⩾1}{inf}{\displaystyle \frac{\alpha \left(I^{\left(m\right)}\right)}{m}}.$$

The following conjecture governing the rate of the growth of the sequence of the initial degrees of the symbolic powers of ideals was stated by Demailly in [23] (p. 101).

Conjecture 1 (Demailly).

Let $U\subset {\mathbb{P}}^N$ be a finite set of points and let $I=I\left(U\right)$ be the homogeneous saturated ideal defining U. Then, for all $m⩾1$,

$$\widehat{\alpha }\left(I\right)⩾{\displaystyle \frac{\alpha \left(I^{\left(m\right)}\right)+N-1}{m+N-1}}.$$

For $m=1$, the Conjecture of Demailly reduces to the following statement which is best known as the Conjecture of Chudnovsky; see [24] (Problem 1).

Conjecture 2 (Chudnovsky).

Keeping the notation of Conjecture 1, the inequality

$$\widehat{\alpha }\left(I\right)⩾{\displaystyle \frac{\alpha \left(I\right)+N-1}{N}}$$

holds for all ideals defining the finite sets of points in${\mathbb{P}}^N$.

Both conjectures have attracted a lot of attention in recent years, with a focus on sets of very general and general points; see, e.g., [8,9,10,11,12,13,25,26,27].

Demailly’s Conjecture for ${\mathbb{P}}^2$ has been proved by Esnault and Viehweg using methods of complex projective geometry; see [28] (Inégalité A). It remains widely open in higher-dimensional projective spaces.

Turning now to the set Z of the 31 points defined in Section 2, the following result is the first main result of this paper.

Theorem 1.

Let Z be the set of points in ${\mathbb{P}}^3$ defined by the ideal $I\left(Z\right)$ in Lemma 1. Then,

$$\widehat{\alpha }\left(I\left(Z\right)\right)=3.$$

Proof. 

It is straightforward to verify that the following polynomials

$$f_1=(x^3-y^3)(z^3-w^3),f_2=(x^3-z^3)(y^3-w^3),f_3=(x^3-w^3)(y^3-z^3)$$

vanish to an order of at least 2 in all points of Z. To see this, it is enough to observe that the zero set of each $f_i$ for $i=1,2,3$ consists of six planes. For each point $P\in Z$, there are at least two planes among those six that contain P. Hence,

$$\widehat{\alpha }\left(I\left(Z\right)\right)⩽{\displaystyle \frac{\alpha \left(I^{\left(2\right)}\right)}{2}}=3.$$

Forthe reverse inequality, we observe that $I\left(Z\right)\subset I\left(X\right)$ implies that

$$\widehat{\alpha }\left(I\left(Z\right)\right)⩾\widehat{\alpha }\left(I\left(X\right)\right).$$

(3)

Since X is a complete intersection of the type $(3,3,3)$, we have $\widehat{\alpha }\left(I\left(X\right)\right)=3$ and, thus, the proof is complete. □

Corollary 1.

The Conjectures of Chudnovsky and Demailly hold for Z.

Proof. 

The Chudnovsky Conjecture holds with $\widehat{\alpha }\left(I\left(Z\right)\right)=3$ and $\alpha \left(I\right)=4$.

Turning to Demailly’s Conjecture, observe to begin with that the symbolic powers of an ideal form a graded family; we have

$$I{\left(Z\right)}^{\left(a\right)}I{\left(Z\right)}^{\left(b\right)}\subset I{\left(Z\right)}^{(a+b)},$$

which implies that

$$\alpha \left(I{\left(Z\right)}^{(a+b)}\right)⩽\alpha \left(I{\left(Z\right)}^{\left(a\right)}\right)+\alpha \left(I{\left(Z\right)}^{\left(b\right)}\right).$$

In particular,

$$\alpha \left(I^{\left(2m\right)}\right)⩽\alpha \left({\left(I^{\left(2\right)}\right)}^m\right)=6m,$$

and the Demailly’s Conjecture is readily verified in this case.

For odd powers, we have

$$\alpha \left(I^{(2m+1)}\right)⩽\alpha ({\left(I^{\left(2\right)}\right)}^m·I\left(Z\right))⩽6m+4,$$

which again implies that Conjecture 1 holds. □

4. Local Positivity

Let X be a complex smooth projective variety. For an ample line bundle L on X, Lazarsfeld, in [22], coined the concept of the local positivity, which is measured by the following invariant introduced by Demailly in [6]; we refer to [29] for details and variations.

Definition 5 (Seshadri constant).

Let V be a smooth projective variety and let L be an ample line bundle on V. Let $P\in V$ be a fixed point and let $f:{\mathrm{Bl}}_{P}V\to V$ be the blow-up of V at P with the exceptional divisor E. The real number

$$\epsilon (V;L,P)=sup\left\{t\in \mathbb{R}:f^{*}L-tEisnef\right\}$$

is the Seshadri constant of L at P.

Thus, $\epsilon (V;L,P)$ is the value of t at which, in the Neron–Severi space of ${\mathrm{Bl}}_{P}X$, the ray $f^{*}L-tE$ hits the boundary of the nef cone. This is analogous to the Waldschmidt constant, which is the value of t at which that ray hits the boundary of the effective cone.

Since the nefness is witnessed by restricting line bundles to curves, there is the following equivalent way to compute the Seshadri constant:

$$\epsilon (V;L,P)=\underset{C\ni P}{inf}{\displaystyle \frac{L.C}{{\mathrm{mult}}_{P}C}},$$

where the infimum is taken over all curves $C\subset V$ passing through P.

More generally, we have the notion of multi-point Seshadri constants.

Definition 6 (Multi-point Seshadri constant).

Let V be a smooth projective variety and let L be an ample line bundle on V. Let $U\subset V$ be a finite set of points. The real number

$$\epsilon (V;L,U)=\underset{C\cap U\ne \varnothing }{inf}{\displaystyle \frac{L.C}{{\sum }_{P\in U}{\mathrm{mult}}_{P}C}},$$

is the multi-point Seshadri constant of L at U.

One of the most intriguing questions in this area is the following.

Question 1 (Rationality problem).

Do there exist irrational multi-point Seshadri constants?

Lazarsfeld pointed out in [22] (Remark 5.1.13) that a negative answer to this question is extremely unlikely. However, it has been 20 years since his book appeared, and to the best of our knowledge, not a single example of an irrational Seshadri constant has been found.

Also, in the case of the 31 points studied here, their multi-point Seshadri constant turns out to be rational. More precisely, we have the following result.

Theorem 2.

Let Z be the set of points in ${\mathbb{P}}^3$ defined by the ideal $I\left(Z\right)$ in Lemma 1. Then,

$$\epsilon ({\mathbb{P}}^3;{\mathcal{O}}_{{\mathbb{P}}^3}\left(1\right),Z)={\displaystyle \frac{1}{4}}.$$

Proof. 

To begin with, note that by taking as C in Definition 6 the line determined by $I=\langle x-y,x-z\rangle$, there are four points of Z on that line, namely,

$$(0:0:0:1),(1:1:1:1),(\epsilon :\epsilon :\epsilon :1),({\epsilon }^2:{\epsilon }^2:{\epsilon }^2:1).$$

It follows that

$$\epsilon ({\mathbb{P}}^3;{\mathcal{O}}_{{\mathbb{P}}^3}\left(1\right),Z)⩽{\displaystyle \frac{1}{4}}.$$

For the opposite direction, assume that there exists an irreducible curve $\Gamma$ such that

$${\displaystyle \frac{deg\Gamma }{{\sum }_{P\in Z}{\mathrm{mult}}_P\Gamma }}<{\displaystyle \frac{1}{4}}.$$

(4)

Let S be the surface of degree 6, which is the union of planes defined by the linear factors of $(x^3-y^3)(z^3-w^3)$, which is $f_1$ in the proof of Theorem 1. This surface has a multiplicity of 3 at the four points of Y and is singular at all points of X with a multiplicity of at least 2.

If $\Gamma$ is not contained in S, we have

$$6deg\Gamma =\Gamma .S⩾\sum _{P\in Z}{\mathrm{mult}}_P\Gamma ·{\mathrm{mult}}_{P}S⩾2\sum _{P\in Z}{\mathrm{mult}}_P\Gamma .$$

Hence,

$${\displaystyle \frac{deg\Gamma }{{\sum }_{P\in Z}{\mathrm{mult}}_P\Gamma }}⩾{\displaystyle \frac{1}{3}},$$

which contradicts (4). If $\Gamma$ is contained in S, we repeat the reasoning with the surface T defined by $(x^3-z^3)(y^3-w^3)$, which is $f_2$ in the proof of Theorem 1.

Thus, $\Gamma$ must be contained in $S\cap T$. However, this intersection consists of 36 lines, each passing through exactly four points of Z, and $\Gamma$ must be one of these lines. But, this contradicts the strict inequality in (4), and the proof is complete. □

Corollary 2.

In the case of the set of the 31 points considered here, Pokora’s question has an affirmative answer, meaning that the multi-point Seshadri constant is computed by lines.

This corollary is particularly interesting because the ambient space of the considered configuration of points is ${\mathbb{P}}^3$, rather than the projective plane, as in the original formulation of the question. In fact, one might expect that, for a set of singular points of an arrangement of planes, the multi-point Seshadri constant would be computed by a plane rather than a line. An easy generalization of [22] (Proposition 5.1.9) from the one-point to the multi-point setting asserts that

$$\epsilon ({\mathbb{P}}^3,{\mathcal{O}}_{{\mathbb{P}}^3}\left(1\right),Z)⩽\underset{S\cap Z\ne \varnothing }{inf}\sqrt{{\displaystyle \frac{degS}{{\sum }_{P\in Z}{\mathrm{mult}}_{P}S}}},$$

(5)

where the infimum is taken over all surfaces containing at least one point of Z.

It is easy to check that at most 10 points from Z are coplanar. Therefore, the infimum taken over the planes provides the upper bound $\sqrt{1/10}\approx 0.3162$, which is significantly worse than the actual value of 0.25.