Let $S$ be a finite set of (reduced) points in the projective plane
and let $I$ be the (saturated) homogeneous ideal of $S$.
Recall that $I^{(m)}$ is the $m$th *symbolic power* of $I$,
consisting of polynomials that vanish to order at least $m$ at each point of $S$
(in characteristic $0$).
Evidently the ordinary power $I^m$ satisfies $I^m \subseteq I^{(m)}$.
This is just the statement that if each one of $F_1,\dotsc,F_m$ vanishes at a point $P$,
then every $(m-1)$th derivative of the product $F_1 \dotsm F_m$ vanishes there also.
In fact if $n \geq m$, then $I^n \subseteq I^{(m)}$.
Conversely, if $I^n \subseteq I^{(m)}$, then $n \geq m$.
So there's a pretty simple classification for when ordinary powers are contained in symbolic powers.

There's no obvious reason that $I^{(m)} \subset I^n$
should ever hold for any $m$ and $n$, beyond the trivial $n=1$, $m \geq 1$.
However, following work of Swanson,
the containment $I^{(2n)} \subseteq I^n$ was shown around 2000 or 2001
by Ein-Lazarsfeld-Smith, using asymptotic multiplier ideals,
and also by Hochster-Huneke, using tight closure methods.
More generally, for ideals of height $h$ (on smooth varieties),
$I^{(hn)} \subseteq I^n$ holds;
for points in the plane the height is $2$.

It's not the case, though, that if $I^{(m)} \subseteq I^n$,
it must be $m \geq 2n$ (or $hn$).
For example, complete intersections have $I^{(m)} \subseteq I^n$ as soon as $m \geq n$.
One can show that if $I^{(cn)} \subseteq I^n$ for all $I$ and all $n$, then it must be $c \geq h$.
But what about small values of $n$, or subleading terms, that is $m = hn + o(n)$?

In particular Huneke asked whether the containment
$I^{(4)} \subseteq I^2$ could be improved to $I^{(3)} \subseteq I^2$.
You can check computationally and it works for lots of examples,
so there's some plausibility.
This was very much a *question*,
but some people (not including Huneke) started calling it "Huneke's conjecture".

Around 2010,
Bocci-Harbourne showed that $I^{(3)} \subseteq I^2$ holds
for points in general position.
That's a dense $G_\delta$: "general position" means it holds on
a Zariski open, dense subset of the $(\mathbb{P}^2)^k$ that parametrizes
sets of $k$ points in the plane (ignoring order and collisions of points).
(On the other hand, this is very much like "all algebraic varieties are smooth".)

But, around 2013, a counterexample was found by
Dumnicki-Szemberg-Tutaj-Gasińka.
It's a collection of $12$ points and easy to verify,
once you know what to try.
It was even a previously known arrangement of points
(a dual Hesse arrangement).

Since then people have found families of counterexamples,
counterexamples in higher dimension,
higher-dimensional counterexamples consisting of positive-dimensional components (instead of points), and so on;
they look for families where $m=cn$ works with $1 \leq c < h$...
You can find literature on this
with keywords like "containment problem for symbolic powers",
resurgence, and Waldschmidt constant.

Why wasn't the counterexample found earlier?
For one thing, it's a bit of a niche subject.
The space of arrangements of $12$ points is $24$-dimensional
(and nobody knew whether $12$ was the right number of points).
And finally, the counterexample is not over the rationals
(it's over $\mathbb{Q}[\omega]$, $\omega$ a cube root of unity),
which means there's an extra step needed to enter it into Macaulay2.
In hindsight that might seem a bit trivial,
but this counterexample wasn't going to be found
by just guessing random field extensions and guessing some points.

Sorry for rambling on.
I think that algebraic geometry must have many versions of this story,
where something was known to hold generally (meaning, on a Zariski open, dense set),
conjectured to hold universally,
but found to have counterexamples.
This particular story is a favorite for me just because it relates to the motivation for my thesis problem.
(I studied the multiplier ideals that appeared in the Ein-Lazarsfeld-Smith proof.)

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