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It is well known that $4$ general points in $\mathbb{P}^2$ are complete intersection of two conics. On the other hand, if $d \geq 3$, $d^2$ general points are not a complete intersection of two curves of degree $d$. More precisely, if $d =3$ there is only one cubic passing through $9$ general points, whereas if $d \geq 4$ there is no curve of degree $d$ passing through $d^2$ general points.

While investigating some questions about factoriality of singular hypersurfaces of $\mathbb{P}^n$, I ran across the following problem, which seems quite natural to state.

Let $d \geq 3$ be a positive integer and let $Q \subset \mathbb{P}^3$ be a subset made of $d^2$ distinct points, with the following property: for a general projection $\pi \colon \mathbb{P}^3 \to \mathbb{P}^2$, the subset $\pi(Q) \subset \mathbb{P}^2$ is the complete intersection of two plane curves of degree $d$.

Is it true that $Q$ itself is contained in a plane (and is the complete intersection of two curves of degree $d$)?

If not, what is a counterexample?

Any answer or reference to the existing literature will be appreciated. Thank you.

EDIT. Dimitri's answer below provides a counterexample given by $d^2$ points on a quadric surface. Are there other configurations of points with the same property? It is possible to classify them up to projective transformations (at least for small values of $d$)?

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up vote 13 down vote accepted

Counterexample. Consider a $Q$ quadric in $\mathbb CP^3$, let $L_1...,L_n$, $M_1,...,M_n$ be lines on $Q$ so that $L_i\cap L_j=\emptyset$, $M_i\cap M_j=\emptyset$, while $L_i$ intersect $M_j$. Take $n^2$ points $L_i\cap M_j$.

Proof. For a generic projection $\pi: \mathbb P^3\to \mathbb P^2$ both collection of lines $L_i$ and $M_j$ project to (reducible) curves of degree $d$ on $\mathbb P^2$. Their intersection are exactly the projections of the collection of points $L_i\cap M_j$. This gives us a pencil of degree $d$ curves, and in a generic situation a generic curve from the pencil will be smooth.

It is interesting to notice that in the case $n=3$ the above construction is rigid, i.e., it produces a unique example up to projective equivalence of $\mathbb P^3$. Are there some further examples of $9$ points in $\mathbb P^3$ having this property? Are they rigid as well?

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Dear Dimitri, thank you very much for your nice answer. Your construction shows that the situation is not so simple as I expected – Francesco Polizzi Jun 9 '11 at 6:29

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