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.



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> 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 $d$. 
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> Is it true that $Q$ itself is contained in a plane (and is the complete intersection of two curves of degree $d$)?
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> If not, what is a counterexample?
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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 $d$)? 



When is a general projection of $d^2$ points in $\mathbb{P}^3$ a complete intersection?
**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? 


