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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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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 statebut I have not been able to solve.

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.

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

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 natural to state but I have not been able to solve.

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.
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