Chudnovsky’s Conjecture and the stable Harbourne-Huneke containment for general points

In the work for providing counterexamples to Hilbert’s $14^{th}$-problem Nagata asked the following question: Take a set of reduced points ${\mathbb{X}}=\{P_{1},\dots,P_{s}\}\subset\mathbb{P}^{2}_{\mathbb{C}}.$ What is the minimal degree $\alpha_{m}({\mathbb{X}})$ of a hypersurface that passes through the given points with multiplicity at least $m$? Nagata conjectured that for at least $10$ general points, $\alpha_{m}({\mathbb{X}})\geqslant m\sqrt{s},\text{ for each }m\geqslant 1$, and proved it for $k^{2}$ many general points (the open condition depends on $m$). The conjecture is still wide open and a vast number of papers in the last few decades are related to this conjecture. Later on, Iarrobino [Iar97] conjectured that $\alpha_{m}({\mathbb{X}})\geqslant m\sqrt[N]{s},$ for sufficiently large number of general points in ${\mathbb{P}}^{N}$. The only known evidence for this conjecture due to Evain [Eva05], for $s=k^{N}$ many general points, when $N\geqslant 3,k\geqslant 3$. These conjectures are equivalent to saying that all the inequalities (for all $m$) hold for (sufficiently many) very general points.

On the other hand, interests for the study of $\alpha_{m}({\mathbb{X}})$ came from other various contexts. We refer interested readers to [CHHVT20] for more information. A more classical motivation of this study is in the context of complex analysis, see [Chu81], [Mor80]. In particular, there have been various studies to get effective lower bounds for $\alpha_{m}({\mathbb{X}})$. Waldschmidt [Wal77] and Skoda [Sko77] proved the inequality $\dfrac{\alpha_{m}({\mathbb{X}})}{m}\geqslant\dfrac{\alpha({\mathbb{X}})}{N}$ for points in ${\mathbb{P}}^{N}_{\mathbb{C}}$ using complex analytic techniques where $\alpha({\mathbb{X}})$ denotes the least degree of a hypersurface that passes through the points at least one time. Chudnovsky[Chu81] improved the bound for points ${\mathbb{P}}^{2}_{{\mathbb{C}}}$, by proving that $\dfrac{\alpha_{m}({\mathbb{X}})}{m}\geqslant\dfrac{\alpha({\mathbb{X}})+1}{2}.$ In the same paper, he conjectured the following inequality for a general set of points in ${\mathbb{P}}^{N}_{\mathbb{C}}$,

All these geometric problems can be re-stated in an algebraic way using the well-celebrated Zariski-Nagata Theorem ([Zariski, Nagata, EisenbudHochster]). More precisely, finding lower bounds for $\alpha_{m}({\mathbb{X}})$ is equivalent to searching for lower bounds for $\alpha\big{(}I^{(m)}\big{)}$, where $I$ is the defining ideal of ${\mathbb{X}}$, $I^{(m)}$ denotes the $m$-th symbolic power of $I$, and $\alpha(J)$ denotes the initial degree of a homogeneous ideal $J$. Thus, Chudnovsky’s conjecture takes the following equivalent format:

The containment problem of symbolic and ordinary powers of ideals is very well-studied (see e.g., [HH13, Sec15, GH17, Gri20, DS21, BGHN22b, BGHN22a, Ngu21, Ngu22b, BFG⁺21].) One of the important applications to study these containment is the fact that the containment would provide lower bounds on the initial degree of the symbolic powers. Consider the following celebrated Theorem by Ein-Lazarsfled-Smith and Hoschter-Huneke:

If $I$ is a defining ideal of points in ${\mathbb{P}}^{N}_{\mathbb{C}}$, then Theorem 1.3 implies $\frac{\alpha\left(I^{(m)}\right)}{m}\geqslant\frac{\alpha(I)}{N},\text{ for all }m\geqslant 1,$ which is the bound proved by Waldschmidt and Skoda. To strengthen the containment, Harbourne-Huneke conjectured that for a homogeneous radical ideal $I\subset{\mathbbm{k}}[{\mathbb{P}}^{N}_{\mathbbm{k}}]$ of big height $N$, one would expect that $I^{(mN)}\subseteq{\mathfrak{m}}^{m(N-1)}I^{m}$ for all $m\geqslant 1$, where ${\mathfrak{m}}=\langle x_{0},x_{1},\dots,x_{N}\rangle$. Chudnovsky’s conjecture follows from stable version of the containment, which has been studied in [BGHN22b].

Previously, the stable Harbourne-Huneke containment $I^{(hr)}\subseteq{\mathfrak{m}}^{r(h-1)}I^{r}$, and hence, Chudnovsky’s conjecture had been shown in the following cases: any set of points in ${\mathbb{P}}^{2}_{\mathbbm{k}}$ [HH13], a general set of points in ${\mathbb{P}}^{3}_{\mathbbm{k}}$ [Dum12, Dum15], a set of at most $N+1$ points in generic position in ${\mathbb{P}}^{N}_{\mathbbm{k}}$ [Dum15], a set of points forming a star configuration [BH10, GHM13]. In addition, Chudnovsky’s conjecture is known for a set of points in ${\mathbb{P}}^{N}_{\mathbbm{k}}$ lying on a quadric [FMX18], and a very general set of points in ${\mathbb{P}}^{N}_{\mathbbm{k}}$ [DTG17, FMX18]. By saying that a property $\mathcal{P}$ holds for a very general set of points in ${\mathbb{P}}^{N}_{\mathbbm{k}}$, we mean that there exist infinitely many open dense subsets $U_{m}$, $m\in{\mathbb{N}}$, of the Hilbert scheme of $s$ points in ${\mathbb{P}}^{N}_{\mathbbm{k}}$ such that the property $\mathcal{P}$ holds for all ${\mathbb{X}}\in\bigcap_{m=1}^{\infty}U_{m}$. If we remove this infinite intersection of open dense subsets and show that there exists one open dense subset $U$ of the Hilbert scheme of $s$ points in ${\mathbb{P}}^{N}_{\mathbbm{k}}$ such that the property $\mathcal{P}$ holds for all ${\mathbb{X}}\in U$, then the property $\mathcal{P}$ holds for a general sets of points. Informally, while very general properties correspond to (intersection of) countable open conditions, general properties correspond to one open condition.

The stable Harbourne-Huneke containment and Chudnovsky’s conjecture was shown to hold for at least $3^{N}$ many general points when $N\geqslant 4$, and the number of points in the results can be reduced to at least $2^{N},\text{when }N\geqslant 9$ in [BGHN22b]. The key idea in the proof is that a stronger containment, namely, $I^{(hr-h)}\subseteq{\mathfrak{m}}^{r(h-1)}I^{r},r\gg 0$, would imply Harbourne-Huneke stable containment. In [BGHN22b], this stronger containment has been proved for a sufficiently large number of generic points, utilizing the important inequality $\widehat{\alpha}(I)>\frac{\operatorname{reg}(I)+N-1}{N}$, where $\widehat{\alpha}(I)$ is the Waldschmidt constant, defined by $\widehat{\alpha}(I):=\lim_{m\rightarrow\infty}\frac{\alpha(I^{(m)})}{m}.$ This required an appropriate lower bound for $\widehat{\alpha}(I)$, but unfortunately, the method in [BGHN22b] could only provide such bounds for sufficiently large (exponential) numbers of generic points, but not for smaller numbers of points.

In this manuscript, we use Cremoma transformation to provide a reduction process to get desired lower bounds for $\widehat{\alpha}(I)$ of the defining ideals of generic points. Our strategy, inspired from the works [Dum09, Dum12, Dum15], is to reduce the study of lower bounds for Waldschmidt constants of defining ideals of generic points to that of a fewer number of generic points. More precisely, we use Cremona transformation as our primary tool to show the following.

As a result of this reduction process combined with a similar approach using specialization as in [BGHN22b], see also [BGHN22a], yields the results on the stable Harbourne-Huneke containment and Chudnovsky’s conjecture for a small number of general points. Combining this and previous results on sufficiently many general points, we are able to complete the picture for all numbers of general points. One key point of the proof is the appropriate lower bound on Waldschmidt constant of generic points.

Note that the Waldschmidt constant for defining ideals of up to $N+3$ generic points are computed in [DHSTG14] and Harbourne-Huneke Containment as well as Chudnovsky’s Conjecture would follow easily, see also [NT19]. Hence, we are interested in ideals defining at least $N+4$ generic points when $N\geqslant 4$. The main result of this paper is the affirmed answer to the stable Harbourne-Huneke Containment and Chudnovsky’s Conjecture for any number of general points in any dimensional projective spaces.

The paper is outlined as follows. Section 2 introduces necessary terminology and notations and recalls some valuable results. In Section 3, we establish Theorems regarding Cremona transformation and obtain lower bounds on the Waldschmidt constant of ideals defining small numbers of generic fat points. In Section 4, we establish the important lower bound for the Waldschmidt constant of generic points. In Section 5, we prove the stable Harbourne-Huneke containment and Chudnovsky’s conjecture for any numbers of general points.

We introduce basic notations and known results that we will be using throughout the paper. We will work with the assumption that $N\geqslant 4$ as both the stable Harbourne-Huneke containment, and Chudnovsky’s conjecture for any sets of general points are known for $N=2$ (see [HH13]) and $N=3$ (see [Dum12, Dum15]). We also use the umbrella assumption that ${\mathbbm{k}}$ is any algebraically closed field. $S={\mathbbm{k}}[{\mathbb{P}}^{N}_{{\mathbbm{k}}}]$ represents the homogeneous coordinate ring of the projective space ${\mathbb{P}}^{N}_{\mathbbm{k}}$. Our work focus on symbolic powers, the Waldschmidt constant, and Cremona transformations, so we define them individually.

We remark here that there is also a notion of symbolic powers in which the set $\operatorname{Min}(I)$ of minimal primes is used in place of the set $\operatorname{Ass}(I)$ of associated primes in the definition. In the context of this paper, for defining ideals of points, or, more generally, ideals with no embedded primes, these two notions of symbolic powers agree. It is well-known that if ${\mathbb{X}}$ is the set $\{P_{1},\dots,P_{s}\}\subseteq{\mathbb{P}}^{N}_{\mathbbm{k}}$ of $s$ many distinct points and let ${\bf p}_{i}\subseteq{\mathbbm{k}}[{\mathbb{P}}^{N}_{\mathbbm{k}}]$ be the defining ideal of $P_{i}$ and $I={\bf p}_{1}\cap\dots\cap{\bf p}_{s}$ is the ideal defining ${\mathbb{X}}$. Then the $m$-th symbolic power is given by,

Using the Waldschmidt constant of defining ideal of set of points in ${\mathbb{P}}^{N}_{\mathbbm{k}}$, Chudnovsky’s conjecture takes the following format.

Let ${\mathbb{X}}=\{P_{1},\dots,P_{s}\}\subset{\mathbb{P}}^{N}_{\mathbbm{k}}$ be a set of points. Then ${\mathcal{L}}_{N}(d;m_{1},\dots,m_{s})$ denotes the linear system of hypersurfaces of degree $d$ passing though the $s$ points $P_{1},\dots,P_{s}$ with multiplicity $m_{1},m_{2},\dots m_{s}$, respectively. In our context, ${\mathcal{L}}_{N}(d;m_{1},\dots,m_{s})=[I(m_{1},\dots m_{s})]_{d}$, the degree $d$-component of the defining ideal.

The following Lemma is due to [Dum09, Theorem 3], see also, [DHSTG14, Lemma B.1.2], which infers how Cremona operations do not alter the linear system up to a certain degree of adjustment. The Lemmas were originally shown for points in general position, but the proof applies for generic points or general points as well. We restate the theorems in our context of defining ideals.

The following Lemmas, due to [Dum09, Theorem 4] and [Dum15, Proposition 10], are very helpful in our reduction process. Our assumption for the set of points is still generic or general.

We also recall some well known results about Waldschmidt constants of defining ideals of small number of points, see also [NT19].

We have mentioned generic and general points many times before. Now we recall some facts about specialization, generic and general points in ${\mathbb{P}}^{N}_{\mathbbm{k}}({\bf z})$. The set of all collections of $s$ not necessarily distinct points in ${\mathbb{P}}^{N}_{\mathbbm{k}}$ is parameterized by the Chow variety $G(1,s,N+1)$ of $0$-cycles of degree $s$ in ${\mathbb{P}}^{N}_{\mathbbm{k}}$ (cf. [GKZ94]). Thus, a property $\mathcal{P}$ is said to hold for a general set of $s$ points in ${\mathbb{P}}^{N}_{\mathbbm{k}}$ if there exists an open dense subset $U\subseteq G(1,s,N+1)$ such that $\mathcal{P}$ holds for any ${\mathbb{X}}\in U$.

Let $(z_{ij})_{1\leqslant i\leqslant s,0\leqslant j\leqslant N}$ be $s(N+1)$ new indeterminates. We shall use ${\bf z}$ and ${\bf a}$ to denote the collections $(z_{ij})_{1\leqslant i\leqslant s,0\leqslant j\leqslant N}$ and $(a_{ij})_{1\leqslant i\leqslant s,0\leqslant j\leqslant N}$, respectively. Let

The set ${\mathbb{X}}({\bf z})$ is often referred to as the set of $s$ generic points in ${\mathbb{P}}^{N}_{{\mathbbm{k}}({\bf z})}$. For any ${\bf a}\in{\mathbb{A}}^{s(N+1)}_{\mathbbm{k}}$, let $P_{i}({\bf a})$ and ${\mathbb{X}}({\bf a})$ be obtained from $P_{i}({\bf z})$ and ${\mathbb{X}}({\bf z})$, respectively, by setting $z_{ij}=a_{ij}$ for all $i,j$. There exists an open dense subset $W_{0}\subseteq{\mathbb{A}}^{s(N+1)}_{\mathbbm{k}}$ such that ${\mathbb{X}}({\bf a})$ is a set of distinct points in ${\mathbb{P}}^{N}_{\mathbbm{k}}$ for all ${\bf a}\in W_{0}$ (and all subsets of $s$ points in ${\mathbb{P}}^{N}_{\mathbbm{k}}$ arise in this way). The following result allows us to focus on open dense subsets of ${\mathbb{A}}^{s(N+1)}_{\mathbbm{k}}$ when discussing general sets of points in ${\mathbb{P}}^{N}_{\mathbbm{k}}$.

We start this section by a consequence of Lemma 2.6 and Lemma 2.7. The following result will be our essential tool to get appropriate lower bounds on the Waldschmidt constant.

The spirit of the proof is the same as [Dum12, Proposition 8].

The following Theorem, inspired by [Dum12, Proposition 10] and [Dum15, Proposition 12], along with the Proposition 3.6 shown below are the main tools in our reduction process.