Nagata-type lower-bound conjectures for Waldschmidt constants of general unions of linear subspaces

The paper 'Linear subspaces, symbolic powers and Nagata type conjectures' (Dumnicki, Harbourne, Szemberg, Tutaj-Gasińska, Advances in Mathematics, 2014) proved that for $s$ general pairwise disjoint $r$-dimensional linear subspaces of $\mathbb P^n$ with $n\ge 2r+1$, the Waldschmidt constant $\gamma(J_{n,r,s})$ — the asymptotic least degree, per unit of multiplicity, of a homogeneous form vanishing to high order along the union — satisfies the upper-bound chain $\gamma(J_{n,r,s})\le e_{n,r,s}\le g_{n,r,s}$, where $g_{n,r,s}$ is the largest real root of an explicit polynomial, and conjectured that equality $\gamma(J_{n,r,s})=g_{n,r,s}$ holds for all sufficiently large $s$. This conjecture simultaneously generalizes Nagata's 1959 conjecture for general points in $\mathbb P^2$ and Iarrobino's extension to higher-dimensional projective spaces. The conjectural lower bound remains open in every regime beyond a few special node counts and arithmetic values of $s$; later work improved lower bounds for lines in $\mathbb P^3$ and for adjacent point problems, and one 2019 paper proves that its own method can never attain the expected value. The problem asks for a proof or refutation of these conjectures in their full stated generality.

Background

Let $k$ be an algebraically closed field and let $\mathbb P^n_k$ be the $n$-dimensional projective space over $k$. For a homogeneous ideal $I$ defining a reduced subscheme $Z\subset\mathbb P^n$ whose components have prime ideals $I(L_i)$, the $m$-th symbolic power $I^{(m)}=\bigcap_i I(L_i)^m$ is the ideal of all homogeneous forms vanishing to order at least $m$ along every component of $Z$. The initial degree $\alpha(I)$ is the least degree of a nonzero form in $I$, and the Waldschmidt constant of $I$ is the limit $\gamma(I)=\lim_{m\to\infty}\alpha(I^{(m)})/m$, which exists by subadditivity. Informally, $\gamma(I)$ measures the asymptotically cheapest degree needed to impose multiplicity $m$ along $Z$, per unit of multiplicity.

Fix integers $n\ge 2r+1$ with $r\ge0$, and let $L_1,\dots,L_s$ be very general (that is, lying outside a countable union of proper closed loci in the configuration space, equivalently generic), pairwise disjoint $r$-dimensional linear subspaces, or $r$-planes, of $\mathbb P^n$, with ideal $J_{n,r,s}=\bigcap_{i=1}^{s}I(L_i)$. For $r=0$ these are general points, and the growth of $\alpha(J_{n,0,s}^{(m)})$ is governed by famous conjectures: Nagata conjectured in 1959 that for $s\ge9$ general points of $\mathbb P^2$ one has $\gamma(J_{2,0,s})=\sqrt{s}$, proving it when $s$ is a perfect square, and Iarrobino extended the prediction to $\mathbb P^n$ for $n\ge3$, where the expected value is $\sqrt[n]{s}$; Evain proved that case when $s$ is an exact $n$-th power. Replacing points by lines in $\mathbb P^3$, Guardo, Harbourne and Van Tuyl posed the analogous question for one-dimensional subspaces.

Dumnicki, Harbourne, Szemberg and Tutaj-Gasińska (2014) unified these questions. Their Theorem 2.4 states that for $n\ge 2r+1$,

$$\gamma(J_{n,r,s})\le e_{n,r,s}\le g_{n,r,s},$$

where $g_{n,r,s}$ is the unique real root at least $1$ of the polynomial

$$\Lambda_{n,r,s}(\tau)=\frac{1}{n!}\Big(\tau^{n}-s\sum_{j=0}^{r}\binom{n}{j}(\tau-1)^{j}\Big),$$

and $e_{n,r,s}$ is the expected Waldschmidt constant determined by the Hilbert polynomial (the polynomial computing the dimension of the space of forms of each degree) $P_{n,r,s,m}$ of the fat subscheme $m(L_1+\cdots+L_s)$, namely $e_{n,r,s}=\inf\{t/m:\ t\ge m\ge1,\ P_{n,r,s,m}(t)>0\}$. For $r=0$ one recovers $g_{n,0,s}=\sqrt[n]{s}$, and for lines in $\mathbb P^3$ one has $g_{3,1,s}\sim\sqrt{3s}$; in general $g_{n,r,s}$ grows like $(s\binom{n}{r})^{1/(n-r)}$. Geometrically, the conjectural equality describes the asymptotic slope of the nef cone — the cone of divisor classes meeting every curve non-negatively — of $\mathbb P^n$ blown up along the $r$-planes. The paper proves $e_{n,r,s}=g_{n,r,s}$ for $s\gg0$ when $r\le1$, and the sharp equality $\gamma(J_{n,1,s})=g_{n,1,s}=n-1$ at the special node count $s=(n-1)^{n-2}$ for general lines in $\mathbb P^n$; it also records that both inequalities can be strict for small $s$. But the lower-bound direction $\gamma(J_{n,r,s})\ge e_{n,r,s}$ is established nowhere, and it is exactly the content of the paper's Conjecture A. Conjecture B asserts $e_{n,r,s}=g_{n,r,s}$ for $s\gg0$ (open for $r\ge2$), and Conjecture C asserts $\gamma(J_{n,r,s})=g_{n,r,s}$ for $s\gg0$; Conjectures A and B together are equivalent to Conjecture C. The open core is thus the lower bound of Conjecture A for every $r$ — including the classical $r=0$ cases of Nagata and Iarrobino — together with Conjecture B for $r\ge2$.

Problem Statement

Decide the Nagata-type conjectures of Dumnicki–Harbourne–Szemberg–Tutaj-Gasińska for general unions of linear subspaces, in their stated generality: for all integers $n\ge2r+1$ with $r\ge0$, determine whether

$$\gamma(J_{n,r,s})=g_{n,r,s}\quad\text{for all sufficiently large }s,$$

where $J_{n,r,s}$ is the ideal of $s$ very general pairwise disjoint $r$-planes in $\mathbb P^n$ and $g_{n,r,s}$ is the largest real root of $\tau^{n}-s\sum_{j=0}^{r}\binom{n}{j}(\tau-1)^{j}$. Equivalently, prove or refute the two constituents: Conjecture A, the lower bound $\gamma(J_{n,r,s})\ge e_{n,r,s}$ for $s\gg0$, where $e_{n,r,s}$ is the expected Waldschmidt constant from the Hilbert polynomial of the corresponding fat subscheme (open for every $r$, and containing the classical Nagata conjecture for points in $\mathbb P^2$ and Iarrobino's generalization to $\mathbb P^n$ as the $r=0$ instances), and Conjecture B, the identity $e_{n,r,s}=g_{n,r,s}$ for $s\gg0$ (open for $r\ge2$; the inequality $e\le g$ is already known). Finitely many small values of $s$ do not decide the conjecture.

References

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2. Marcin Dumnicki, Mohammad Zaman Fashami, Justyna Szpond, Halszka Tutaj-Gasińska, 'Lower bounds for Waldschmidt constants of generic lines in $\mathbb{P}^3$ and a Chudnovsky-type theorem', Mediterranean Journal of Mathematics 16 (2019), article 53, DOI 10.1007/s00009-019-1328-8, arXiv:1803.02387, https://arxiv.org/abs/1803.02387
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