Exact Waldschmidt constants for singular loci of complex reflection arrangements: the Klein configuration

For a finite set of points $Z$ in the complex projective plane with homogeneous ideal $I$, the Waldschmidt constant $\widehat{\alpha}(I)=\lim_{m\to\infty}\alpha(I^{(m)})/m$ is the asymptotic slope of the least degree of a form vanishing to order $m$ at each point of $Z$; it measures how far the configuration sits from the generic behavior predicted by Nagata's still-open conjecture ($\widehat{\alpha}=\sqrt{s}$ for $s\geq 10$ very general points). Highly symmetric configurations arise as singular loci of line arrangements attached to complex reflection groups in the Shephard–Todd classification. Most named plane cases are now settled — the Fermat families, the Wiman configuration ($27/2$), and the icosahedral 31-point configuration ($11/2$) — but the exact constant for the Klein configuration of 49 points (the 21 quadruple and 28 triple points of the 21-line arrangement of the group $G_{24}\simeq PSL(2,7)$) remains unknown: only $661/102\leq\widehat{\alpha}\leq 13/2$ is proved, with $\widehat{\alpha}=13/2$ conjectured via nefness of a specific divisor on the blown-up plane. The problem asks for the exact Waldschmidt constants of the still-unresolved reflection-arrangement singular loci, the Klein configuration above all.

Background

Let $Z=\{p_1,\dots,p_s\}\subset\mathbb{P}^2=\mathbb{P}^2_{\mathbb{C}}$ be a finite set of points with homogeneous ideal $I(Z)\subset S=\mathbb{C}[x,y,z]$. The $m$-th symbolic power of $I(Z)$ is $I(Z)^{(m)}=\bigcap_i I(p_i)^m$, the ideal of all forms vanishing to order at least $m$ at every point of $Z$. Writing $\alpha(J)$ for the least degree of a nonzero element of a homogeneous ideal $J$, the initial sequence of the configuration is $(\alpha(I^{(m)}))_{m\geq 1}$, and the Waldschmidt constant is its asymptotic slope

$\widehat{\alpha}(I)=\lim_{m\to\infty}\frac{\alpha(I^{(m)})}{m}=\inf_{m\geq 1}\frac{\alpha(I^{(m)})}{m}.$

Always $1\leq\widehat{\alpha}(I)\leq\sqrt{s}$, and the upper equality is conjectured for very general points (points avoiding a countable union of proper closed conditions): Nagata's conjecture asserts $\widehat{\alpha}=\sqrt{s}$ for $s\geq 10$ very general points in $\mathbb{P}^2$, equivalently that any curve of degree $d$ with multiplicities $m_i$ at the points satisfies $d\geq\frac{1}{\sqrt{s}}\sum_i m_i$. Nagata's conjecture remains open, proved only in special cases such as $s$ a perfect square (Ciliberto–Miranda). For special, non-general configurations one expects and observes $\widehat{\alpha}<\sqrt{s}$, so the exact value quantifies how special the configuration is; equivalently, $1/\widehat{\alpha}$ is a multi-point Seshadri constant, a measure of local positivity of the hyperplane bundle.

The configurations studied here come from line arrangements with large symmetry groups. A (line) arrangement in $\mathbb{P}^2$ is a finite union of lines, and its singular locus is the set of points where at least two of the lines meet. A complex reflection group is a finite subgroup of $GL(V)$, $\dim V=n+1$, generated by pseudoreflections — elements fixing a hyperplane pointwise; these groups were classified by Shephard and Todd into finitely many exceptional groups plus the imprimitive families $G(m,p,n)$. The reflecting hyperplanes of such a group projectivize to a reflection arrangement in $\mathbb{P}^n$ whose singular locus is a highly symmetric point configuration. In the plane the relevant configurations include: the Fermat configurations of $m^2+3$ points from the monomial groups $G(m,m,3)$ (the case $m=3$ being the dual Hesse configuration of 12 points); the Klein configuration of 49 points — the 21 quadruple and 28 triple intersection points of the 21 lines fixed by the involutions of $G_{24}\simeq PSL(2,7)\subset PGL_3(\mathbb{C})$, the order-168 simple group of the Klein quartic; the Wiman configuration of 201 points from $G_{27}\simeq A_6$ (45 lines meeting in 36 quintuple, 45 quadruple, and 120 triple points); and the icosahedral configuration of 31 points — 6 quintuple, 10 triple, and 15 double points of the 15 lines coming from the full symmetry group $A_5\times\mathbb{Z}_2$ of the icosahedron.

The standard method for computing $\widehat{\alpha}$ for such a configuration is geometric. One blows up $\mathbb{P}^2$ at the points of $Z$; the Picard group of the resulting surface $X$ is generated by the pullback $H$ of a line and the exceptional divisors over the points, and for an arrangement with all singularities of multiplicities $k$ in single orbits under the group one writes $E_k$ for the sum of exceptional divisors over the $k$-uple points. A divisor on a surface is nef if it intersects every effective curve non-negatively. Upper bounds for $\widehat{\alpha}$ come from explicit curves of low degree with high multiplicities at the points; since the group $G$ acts on $X$, it suffices to use $G$-invariant curves, whose defining equations lie in the invariant ring $S^G$ — a polynomial algebra $\mathbb{C}[\Phi_a,\Phi_b,\Phi_c]$ on three basic invariants by the Shephard–Todd–Chevalley theorem. Lower bounds come from proving that a suitable $G$-invariant divisor $D=dH-\sum m_k E_k$ is nef: an effective curve $F=\beta H-m\sum_k E_k$ (a curve of degree $\beta$ with multiplicity at least $m$ at every point of $Z$) must satisfy $F\cdot D\geq 0$, which bounds the ratio $\beta/m$ from below, and taking the infimum over $m$ bounds $\widehat{\alpha}$ from below. Proving nefness reduces, through the group action, to classifying the $G$-invariant, $G$-irreducible curves (curves whose components form a single orbit) of negative self-intersection on $X$ — and this classification is exactly where the unresolved difficulty lies for the Klein configuration, where this classification is currently known only in bounded degree (degree $200$ in Bauer–Di Rocco–Harbourne–Huizenga–Seceleanu–Szemberg), and a $G$-invariant curve of negative self-intersection meeting $D$ negatively would give a smaller value of $\widehat{\alpha}(I_K)$. These configurations are also the standard source of counterexamples in the containment problem between symbolic and ordinary powers of ideals (e.g. $I^{(3)}\not\subseteq I^2$), which is one of the main reasons their asymptotic invariants are actively sought.

Problem Statement

Determine the exact Waldschmidt constant of the ideal of the singular locus of the Klein configuration $K$, the flagship open instance treated below. Proving or disproving $\widehat{\alpha}(I_K)=13/2$ is a complete resolution of the task.

The flagship open instance is the Klein configuration $K$: the 49 singular points (21 quadruple and 28 triple points) of the arrangement of 21 lines in $\mathbb{P}^2$ arising from the exceptional reflection group $G_{24}\simeq PSL(2,7)\subset PGL_3(\mathbb{C})$. Prove or refute that $\widehat{\alpha}(I_K)=13/2$; currently only

$\frac{661}{102}\approx 6.480 \leq \widehat{\alpha}(I_K) \leq \frac{13}{2}$

is known. The conjectured value would follow from nefness of the divisor $D=28H-2E_4-5E_3$ on the blowup $X_K$ of $\mathbb{P}^2$ at the 49 points, where $H$ is the pullback of a line, $E_4$ and $E_3$ are the sums of the exceptional divisors over the quadruple and triple points (Bauer–Di Rocco–Harbourne–Huizenga–Seceleanu–Szemberg, Conjecture 5.8).

Beyond Klein, the same question may be asked of the remaining singular loci of reflection arrangements whose constants are not yet known — in the plane, 'most but not all' cases are settled per Calvo–Huizenga–Szemberg (2025), and the program naturally extends to configurations in $\mathbb{P}^3$ and higher, where the identification of divisors with curves fails and the existing surface-based method does not transfer; these further loci are contextual extension, not a required deliverable. A satisfactory resolution of the task therefore consists of proving or disproving $\widehat{\alpha}(I_K)=13/2$ for the Klein configuration (equivalently, resolving Conjecture 5.8 of Bauer–Di Rocco–Harbourne–Huizenga–Seceleanu–Szemberg, or Harbourne's Open Problem 1.3.17). The generic benchmark $\widehat{\alpha}=\sqrt{s}$ of Nagata's conjecture supplies the motivating comparison but is not itself part of the deliverable.

References

1. S. Calvo, The icosahedral line configuration and Waldschmidt constants, Journal of Pure and Applied Algebra 228 (2024), no. 6, Article 107563, DOI 10.1016/j.jpaa.2023.107563, arXiv:2209.01499, https://arxiv.org/abs/2209.01499
2. T. Bauer, S. Di Rocco, B. Harbourne, J. Huizenga, A. Seceleanu, T. Szemberg, Negative curves on symmetric blowups of the projective plane, resurgences, and Waldschmidt constants, International Mathematics Research Notices 2019 (2019), no. 24, 7459-7514, DOI 10.1093/imrn/rnx329, arXiv:1609.08648, https://arxiv.org/abs/1609.08648
3. S. Calvo, J. Huizenga, T. Szemberg, Waldschmidt constants of symmetric sets of points in $\mathbb{P}^3$, Journal of Pure and Applied Algebra 230 (2026), no. 5, Article 108260, DOI 10.1016/j.jpaa.2026.108260, arXiv:2507.11888, https://arxiv.org/abs/2507.11888
4. B. Drabkin, A. Seceleanu, Singular loci of reflection arrangements and the containment problem, Mathematische Zeitschrift 299 (2021), no. 1-2, 867-895, DOI 10.1007/s00209-021-02701-1, arXiv:2002.05353, https://arxiv.org/abs/2002.05353
5. Thai Thanh Nguyen, Initial degree of symbolic powers of ideals of Fermat configurations of points, Rocky Mountain Journal of Mathematics 53 (2023), no. 3, 859 ff., DOI 10.1216/rmj.2023.53.859, arXiv:2101.12308, https://arxiv.org/abs/2101.12308
6. M. Nagata, On the 14-th problem of Hilbert, American Journal of Mathematics 81 (1959), no. 3, 766-772, DOI 10.2307/2372927, https://doi.org/10.2307/2372927
7. C. Ciliberto, R. Miranda, Nagata's conjecture for a square or nearly-square number of points, Ricerche di Matematica 55 (2006), no. 1, 71-78, DOI 10.1007/s11587-006-0005-y, https://doi.org/10.1007/s11587-006-0005-y
8. L. Farnik, J. Gwozdziewicz, B. Hejmej, M. Lampa-Baczynska, G. Malara, J. Szpond, Initial sequences and Waldschmidt constants of planar point configurations, International Journal of Algebra and Computation 27 (2017), no. 6, 717-729, DOI 10.1142/S0218196717500345, arXiv:1607.01031, https://arxiv.org/abs/1607.01031
9. S. Bisui, T. T. Nguyen, Lower bounds for Waldschmidt constants and Demailly's conjecture for general and very general points, Collectanea Mathematica 77 (2025), 483-500, DOI 10.1007/s13348-025-00477-9, arXiv:2401.11297, https://arxiv.org/abs/2401.11297
10. B. Harbourne, Asymptotics of linear systems, with connections to line arrangements, arXiv:1705.09946, https://arxiv.org/abs/1705.09946
11. G. C. Shephard, J. A. Todd, Finite unitary reflection groups, Canadian Journal of Mathematics 6 (1954), 274-304, DOI 10.4153/cjm-1954-028-3, https://doi.org/10.4153/cjm-1954-028-3