\begin{abstract}
We prove the Ideal Generation Conjecture for general reduced points
in projective space over an algebraically closed field of
characteristic zero. Equivalently, if the ideal of the points has
initial degree $D$, multiplication by linear forms from degree $D$
to degree $D+1$ has maximal rank. The proof combines hyperplane
specialization with a construction using two transverse hyperplanes.
In the latter construction, motion of their common trace gives
the required rank. A uniform induction for all degrees $D\ge4$
reduces the theorem to explicit quadratic and cubic constructions.
\end{abstract}
\maketitle

\section{Introduction}

Let $Z$ be a general set of $p$ points in $\mathbb P^{n-1}$, and let
$I_Z\subset S=k[x_1,\ldots,x_n]$ be its homogeneous ideal.
The Hilbert function of $S/I_Z$ is
$\min\{p,\dim_k S_j\}$ in degree $j$.
Determining the minimal generators of $I_Z$ asks a further question:
how much of its first nonzero component is needed to generate the
next one? The Ideal Generation Conjecture predicts that
multiplication by linear forms has the largest rank allowed by
these dimensions.

The minimal generators of ideals of general points were studied by
Geramita and Orecchia \cite{GO82} and Geramita and Maroscia \cite{GM84}.
For a recent formulation of the conjecture and its relation to
ideals generated by a single homogeneous component, see
Gesmundo, Kayser, and Telen \cite[Sections~2.2--2.3]{GKT25}.
We prove the following statement.

\begin{theorem}\label{thm:igc}
Let $k$ be an algebraically closed field of characteristic zero,
let $n\ge2$ and $D\ge2$, and put
\[
 N_j(n)=\binom{n+j-1}{j}.
\]
For every integer $p$ satisfying
\[
 N_{D-1}(n)\le p<N_D(n),
\]
a general reduced set $Z$ of $p$ points in $\mathbb P^{n-1}$ has
maximal-rank multiplication
\[
 (I_Z)_D\otimes_k S_1\longrightarrow (I_Z)_{D+1}.
\]
\end{theorem}

In particular, put $r=N_D(n)-p$. The ideal has $r$ minimal
generators in degree $D$ and
\[
 \max\left\{0,(n-1)\left(\frac{N_D(n)}{D+1}-r\right)\right\}
\]
in degree $D+1$. There are no generators in higher degrees:
the scalar postulation of general points gives
$(D+1)$-regularity of their ideal sheaf.
For a set of points spanning a proper linear subspace, its linear
equations can first be split off and the theorem applied in its span.

The proof treats ideal generation as interpolation for
$\Omega_{\mathbb P^{n-1}}(D+1)$ via the Euler sequence.
This places the argument within the vectorial Horace method;
compare Hirschowitz and Simpson \cite{HS96} and
Lauze \cite[Section~3]{Lauze95}.
All the specialization arguments used here are developed explicitly.

For every $D\ge4$, we deduce the degree-$D$ statement from the
statements in degrees $D-1$ and $D-2$. At fixed degree, hyperplane
specialization, including a two-level step for fractional critical
counts, covers $n\le D(D-1)+1$.
For larger $n$, we use two hyperplanes with common trace
$\mathbb P^{n-3}$. Scalar interpolation identifies both the kernel
and the cokernel of a specialized evaluation map. Moving the trace
in its two normal directions then supplies a maximal-rank first
variation. When the trace itself has a kernel, one residual point
with a free projection detects it.

The quadratic and cubic cases start this induction. Their explicit
constructions and finite determinant calculations are given in
Appendices~\ref{sec:quadratic}--\ref{sec:cubic-determinants}.
Degree four is the first application of the same induction:
the hyperplane steps reach thirteen variables, and the two-normal
steps cover every larger dimension.
Section~\ref{sec:induction} assembles the transfers into the proof.