\section{Ideal generation and cotangent interpolation}\label{sec:preliminaries}

Throughout, $k$ is an algebraically closed field of characteristic zero.
We write
\[
 S=k[x_1,\ldots,x_n],\qquad
 N_j(n)=\dim_k S_j=\binom{n+j-1}{j},\qquad
 E(n,D)=\frac{N_D(n)}{D+1}.
\]
If $Z$ is a finite reduced set of points, $I_Z$ denotes its
homogeneous ideal and $\mathcal I_Z$ its ideal sheaf.
A statement about general ordered points means that it holds on a
nonempty Zariski open subset of the space of distinct ordered points.
All the conditions considered below are unchanged by reordering.

\subsection{Scalar interpolation and the two critical counts}

General points impose independent conditions on forms: for every $j$,
\begin{equation}\label{eq:scalar-postulation}
 \dim_k(I_Z)_j=\max\{0,N_j(n)-|Z|\}.
\end{equation}
Indeed, evaluation functionals at points span $S_j^*$, since a form
vanishing at every point is zero. Successively choosing independent
functionals gives a configuration of the required rank, and the
nonvanishing of the corresponding minor gives the assertion for
general points. Finitely many such rank conditions hold simultaneously.

Suppose
\begin{equation}\label{eq:initial-cell}
 N_{D-1}(n)\le p<N_D(n),\qquad r=N_D(n)-p .
\end{equation}
For points with the scalar postulation in degrees $D-1,D,D+1$, the
multiplication map
\begin{equation}\label{eq:multiplication}
 \mu_Z:(I_Z)_D\otimes_k S_1\longrightarrow (I_Z)_{D+1}
\end{equation}
has source dimension $nr$ and target dimension $N_{D+1}(n)-p$.
Their difference is
\begin{equation}\label{eq:euler-index}
 nr-\bigl(N_{D+1}(n)-p\bigr)
   =(n-1)\bigl(r-E(n,D)\bigr).
\end{equation}
Thus the critical ideal dimensions are $\lfloor E(n,D)\rfloor$
and $\lceil E(n,D)\rceil$.
They belong to the range in \eqref{eq:initial-cell}: the upper bound
follows from
\[
 E(n,D)\le N_D(n)-N_{D-1}(n)=N_D(n-1),
\]
which is equivalent to $D(n-2)\ge0$.

\begin{lemma}[Reduction to the critical counts]\label{lem:critical-counts}
To prove maximal rank of \eqref{eq:multiplication} for general points
throughout \eqref{eq:initial-cell}, it is enough to prove injectivity
when $r=\lfloor E(n,D)\rfloor$ and surjectivity when
$r=\lceil E(n,D)\rceil$.
\end{lemma}

\begin{proof}
Adding points replaces $(I_Z)_D$ by a subspace, so an injective
multiplication map remains injective.
For surjectivity, let $Y\subset Z$ and assume that evaluation of
degree-$D$ forms on $Z$ is onto. Choose a linear form $\ell$ nonzero
at every point of $Z$ and use the affine trivializations $\ell=1$.
Given $g\in(I_Y)_{D+1}$, scalar interpolation supplies $f\in S_D$
which is zero on $Y$ and agrees with $g$ on $Z\setminus Y$.
Then $f\in(I_Y)_D$ and $g-\ell f\in(I_Z)_{D+1}$.
If $\mu_Z$ is onto, this difference is a sum of linear multiples
of elements of $(I_Z)_D\subset(I_Y)_D$, proving that $\mu_Y$ is onto.
The relevant evaluation kernels form vector bundles on the scalar
postulation open sets; maximal rank of multiplication is an open
condition there. Hence each configuration constructed in this way
proves the assertion for general points of its cardinality.
\end{proof}

We write $\operatorname{IGC}(n,D)$ for the assertion of this lemma
for all the point counts in \eqref{eq:initial-cell}.
We write $\operatorname{Inj}(n,D,r)$ for its injective assertion
at ideal dimension $r$.

\subsection{The Euler sequence}

Let
\[
 \mathcal E_{n,D}=\Omega_{\mathbb P^{n-1}}(D+1).
\]
The Euler sequence identifies its sections with polynomial relations:
\begin{equation}\label{eq:euler-sections}
 H^0(\mathcal E_{n,D})
 =\left\{(F_1,\ldots,F_n)\in S_D^n:
               \sum_{i=1}^n x_iF_i=0\right\}.
\end{equation}
Its rank is $n-1$ and its space of sections has dimension
$nN_D(n)-N_{D+1}(n)$.

\begin{lemma}[Euler evaluation]\label{lem:euler-evaluation}
Suppose $Z$ imposes independent conditions in degrees $D$ and $D+1$.
Then
\[
 \operatorname{ev}_Z:H^0(\mathcal E_{n,D})
       \longrightarrow H^0(\mathcal E_{n,D}|_Z)
\]
and $\mu_Z$ have naturally isomorphic kernels and cokernels.
In particular, they have maximal rank simultaneously.
\end{lemma}

\begin{proof}
Compare the Euler sequence on global sections with its restriction
to $Z$. The two rows
\[
\begin{array}{ccccccccc}
0&\longrightarrow&H^0(\mathcal E_{n,D})&
 \longrightarrow&S_D^n&\longrightarrow&S_{D+1}&\longrightarrow&0\\
&&\downarrow&&\downarrow&&\downarrow\\
0&\longrightarrow&H^0(\mathcal E_{n,D}|_Z)&
 \longrightarrow&H^0(\mathcal O_Z(D))^n&
 \longrightarrow&H^0(\mathcal O_Z(D+1))&\longrightarrow&0
\end{array}
\]
are exact. The last arrow of the bottom row is onto because the
homogeneous coordinates do not all vanish at any point.
The last two vertical arrows are onto by scalar postulation.
The snake lemma gives
\[
0\longrightarrow\ker\operatorname{ev}_Z
 \longrightarrow (I_Z)_D^n
 \xrightarrow{\mu_Z}(I_Z)_{D+1}
 \longrightarrow\operatorname{coker}\operatorname{ev}_Z
 \longrightarrow0,
\]
as required.
\end{proof}

\subsection{Rank under deformation}

All configurations used below are actual point configurations, and
all limiting calculations concern their evaluation maps.
We record the elementary matrix principle used in these calculations.

\begin{lemma}[First variation of rank]\label{lem:first-variation}
Let $A(\varepsilon):V\to W$ be a matrix regular at $\varepsilon=0$,
and put $K=\ker A(0)$ and $Q=\operatorname{coker}A(0)$.
The coefficient of $\varepsilon$ induces a map
$J:K\to Q$. If $J$ has maximal rank, then $A(\varepsilon)$ has
maximal rank for general $\varepsilon$.
\end{lemma}

\begin{proof}
Choose complements to $K$ and to $\operatorname{im}A(0)$.
The block from the first complement to $\operatorname{im}A(0)$
is invertible at zero. Eliminating it leaves a block
$\varepsilon J+O(\varepsilon^2)$ from $K$ to $Q$.
A nonzero maximal minor of $J$ is the leading coefficient
of the corresponding minor of this block.
Together with the invertible block it gives a minor of size
$\rank A(0)+\min\{\dim K,\dim Q\}=\min\{\dim V,\dim W\}$.
\end{proof}

When several open rank conditions are imposed on a single
irreducible parameter space, it is enough to exhibit a point for
each condition separately. Their nonempty open loci then intersect.
In each application below, the parameter space and the projection
to any auxiliary point configuration are specified explicitly.