Effective Cone Constants

• Effective cone constants are explicit numerical thresholds that characterize the asymptotic behavior of convex cones and divisorial invariants.
• They are derived using detailed integral representations with Beta and Gamma functions to compute key metrics like mean face counts and intrinsic volumes.
• In algebraic geometry, these constants inform the structure of effective cones and Seshadri thresholds, impacting wall–chamber decompositions and moduli space models.

An effective cone constant refers to a limiting or explicit constant governing the asymptotic geometric, combinatorial, or algebro-geometric structure of convex cones or their associated invariants, appearing most notably in random convex geometric settings, algebraic geometry (notably moduli of sheaves), and the local positivity theory of divisors. These constants encode either expectation values (mean face counts, Grassmann angles, intrinsic volumes), sharp lower bounds (area-minimizing cones), or thresholds corresponding to structural changes in divisorial geometry (Seshadri constants, extremal rays in effective cones).

1. Constants for Convex Cones Generated by Random Points

Let $C_n \subset \mathbb{R}^{d+1}$ be the convex polyhedral cone generated by $n$ i.i.d. points $U_1,\ldots,U_n$ on the upper half-sphere $$\mathbb{S}_+^d = \{ x\in\mathbb{R}^{d+1}:\|x\|=1, x_0\ge0 \}$$. The $k$-face count $f_k(C_n)$ is of central interest. As $n\to\infty$,

• The (rescaled) random cone $C_n$ converges weakly to a random cone determined by the convex hull of a Poisson point process $\Pi_{d,1}(2)$ of intensity $$\nu(\mathrm{d}x) = \tfrac{2}{\omega_{d+1}} \|x\|^{-(d+1)} \mathrm{d}x$$ on $\mathbb{R}^d \setminus \{0\}$.
• The expected $k$-face count admits a limiting value

$$c_{d,k} := \lim_{n\to\infty} \mathbb{E}[f_k(C_n)] = \mathbb{E}[f_k(\Pi_{d,1}(2))] = \frac{2}{k!} B_{k,d},$$

where $B_{k,d}$ is an explicit Poisson-origin integral involving the intersection structure of random convex hulls.

The facet constant for $k=d$ is available in closed form:

$$c_{d,d} = \lim_{n\to\infty} \mathbb{E} f_d(C_n) = 2^{-d} d! \kappa_d^2, \quad \kappa_d = \frac{\pi^{d/2}}{\Gamma(\frac{d}{2}+1)}$$

where $\kappa_d$ is the volume of the $d$-dimensional unit ball. The vertex constant $c_{d,1}$ is available in terms of hypergeometric or Beta/Gamma integrals (Kabluchko et al., 2018).

2. Geometric Interpretation via Poisson Point Processes

The limiting behavior of the cone is understood by mapping points $U_i$ via the affine chart $P(x_0, x_1, ..., x_d) = (x_1/x_0, ..., x_d/x_0)$, yielding a scaled random point process in $\mathbb{R}^d$, which converges in distribution to $\Pi_{d,1}(2)$. In this framework, geometric functionals of $C_n$—including $f$-vectors, Grassmann angles, and conic intrinsic volumes—are determined by corresponding functionals of the convex hull of $\Pi_{d,1}(2)$. This Poisson-based approach enables direct calculation of all limiting effective cone constants relevant to asymptotic face/volume statistics, Gaussian-projection intrinsic volumes, and rates for Grassmann angles and mean projection volumes (Kabluchko et al., 2018).

3. Explicit Constant Formulas and Integral Representations

For general exponents and power-law intensity, explicit closed-form expressions and master integral formulas for limiting constants are established. For instance, for facets of the convex hull of a Poisson process with intensity $c\,\|x\|^{-(d+\gamma)}$, one has

$$\mathbb{E} f_{d-1}(\Pi_{d,\gamma}(c)) = \frac{2}{d}\gamma^{d-1} \pi^{(d-1)/2} \frac{\Gamma(\tfrac{\gamma d + 1}{2})}{\Gamma(\tfrac{\gamma d}{2})} \left( \frac{\Gamma(\tfrac{\gamma}{2})}{\Gamma(\tfrac{\gamma+1}{2})} \right)^d,$$

independent of $c$. Specializations recover the constants relevant to cones generated by half-sphere points ($\gamma=1, c=2$).

For intrinsic volumes $v_\ell(C_n)$ ($\ell=0,...,d$), asymptotic constants are

$$\lim_{n\to\infty} n^{d-\ell} \mathbb{E} v_\ell(C_n) = B_{d-\ell,d}$$

with $B_{*,*}$ as above. The $T$-functional method provides a unified framework generating all such constants in terms of Beta and Gamma integrals (Kabluchko et al., 2018).

4. Constants Arising in Algebraic and Birational Geometry

In the context of projective varieties $X$, the effective cone $\mathrm{Eff}(X)\subseteq N^1(X)$ is generated by (classes of) effective Cartier divisors. For moduli spaces $M(\xi)$ of Gieseker semistable sheaves on $P = \mathbb{P}^1 \times \mathbb{P}^1$, $M(\xi)$ is a rational polyhedral Mori dream space: both $\mathrm{Eff}(M(\xi))$ and the nef cone are finite rational polyhedral cones. Extremal rays of $\mathrm{Eff}(M(\xi))$ are explicitly generated by Brill–Noether divisors associated to stable vector bundles $V$ orthogonal to the fixed Chern character, with divisor classes determined via associated dual curves. The explicit determination of generators and the corresponding facet inequalities—parametrized by effective cone constants such as $X_{i,j}$, $Y_{a,b}$—enable complete, finite, polyhedral descriptions for $$\mathrm{Eff}(\mathrm{Hilb}^n(\mathbb{P}^1 \times \mathbb{P}^1))$$, with a closed table for all $n \leq 16$ (Ryan, 2016).

A consolidated description is:

• The effective cone is defined by explicit linear inequalities in divisor coordinates $(a, b, c)$.
• Extremal rays and walls correspond to vanishing intersection numbers for covering (moving) curves, giving a finite wall–chamber decomposition of the cone.
• These effective cone constants play a crucial role in birational models and wall-crossing phenomena in moduli theory (Ryan, 2016).

5. Seshadri Constants as Effective Cone Constants

Seshadri constants $\varepsilon(L;x)$, defined as

$$\varepsilon(L;x) = \inf_{C \ni x} \frac{L\cdot C}{\mathrm{mult}_x(C)},$$

quantify local positivity of nef line bundles and frequently arise as critical effective cone constants. Recent work connects them directly to the walls of nef, movable, and effective cones of Hilbert and nested Hilbert schemes. For very general K3 surfaces of Picard rank one, the wall of the movable cone defined by

$$H^{[3]} - \frac{\varepsilon(H)}{2} B \in \partial \mathrm{Mov}(X^{[3]}),$$

matches the Seshadri constant, and similar relations hold generally (Baltes, 2024).

In the nested Hilbert scheme $X^{[r, r+1]}$, the boundary rays of the nef and effective cones encode, respectively, the infimum and supremum multi-point Seshadri constants:

• Nef cone: $$\varepsilon_{\inf}(H, r) = \sup\{ \lambda > 0 \mid H_\Delta - \lambda E + A \textrm{ ample for some } A \}$$,
• Effective cone: $$H^2/(r\,\varepsilon_{\sup}(H, r)) = \sup \{ \lambda > 0 \mid H_\Delta-\lambda E + A \textrm{ is }\mathbb{Q}\textrm{-effective} \}$$.

In specific cases, such as $X = \mathbb{P}^2$, explicit constants like $\varepsilon_{\inf}(H, r) = 1/r$ and $\varepsilon_{\sup}(H, r) = 1/r$ are recovered. For general K3 surfaces, new explicit lower bounds are derived, e.g.,

$$\varepsilon(H) \geq \frac{2 H^2 n}{(a+1)H^2 + n(n+1) + 2}$$

for appropriate $n$, extending older bounds (Baltes, 2024).

6. Effective Cone Constants in Seshadri Theory on Projective Bundles

In the context of projective bundle products $X = \mathbb{P}(E_1) \times_C \mathbb{P}(E_2)$ (over a curve $C$), the nef cone is given by

$$\mathrm{Nef}(X) = \{ aT_1 + bT_2 + cF \mid a,b,c\geq0 \}$$

where $T_i$ is defined via the minimal Harder–Narasimhan slope of $E_i$, and $F$ is the fibre class. The sharp lower bound for the Seshadri constant for any nef line bundle $L = aT_1 + bT_2 + cF$ is precisely

$\varepsilon(L, x) \geq \min\{a,b,c\}$

and is always realized for some $x$ by suitable extremal, effective curves (Karmakar et al., 2018). Thus, the sets of coefficients $\{a,b,c\}$ in the nef cone serve as effective cone constants for Seshadri-type invariants in this context.

7. Sharp Lower Bounds for Area-Minimizing Cones

For minimal surfaces and area-minimizing hypercones $C \subset \mathbb{R}^{n+1}$ with an isolated singularity at the origin, the density at $0$,

$$\Theta(C) = \lim_{r \to 0} \frac{\mathrm{Vol}_n(C \cap B(0,r))}{\omega_n r^n}$$

is subject to universal sharp lower bounds. If $C$ is topologically nontrivial,

$\Theta(C) > \sqrt{2}$

and, under further topological constraints (nontrivial $k$-th homotopy),

$$\Theta(C) \geq d_k = \frac{1}{(4\pi)^{k/2} \, \omega_k}$$

where $d_k$ is the Gaussian density of a shrinking $k$-sphere in $\mathbb{R}^{k+1}$. Simons' cones demonstrate the sharpness of these constants. These effective cone constants serve as critical regularity barriers and minimal thresholds within geometric measure theory and mean curvature flow (Ilmanen et al., 2010).

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References:

• (Kabluchko et al., 2018) Cones generated by random points on half-spheres and convex hulls of Poisson point processes
• (Ryan, 2016) The Effective Cone of Moduli Spaces of Sheaves on a Smooth Quadric Surface
• (Baltes, 2024) Hilbert Schemes and Seshadri Constants
• (Karmakar et al., 2018) Nef cone and Seshadri constants on products of projective bundles over curves
• (Ilmanen et al., 2010) Sharp Lower Bounds on Density of Area-Minimizing Cones