When is a general projection of $d^2$ points in $\mathbb{P}^3$ a complete intersection? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T02:42:27Z http://mathoverflow.net/feeds/question/67265 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/67265/when-is-a-general-projection-of-d2-points-in-mathbbp3-a-complete-interse When is a general projection of $d^2$ points in $\mathbb{P}^3$ a complete intersection? Francesco Polizzi 2011-06-08T15:09:19Z 2011-06-09T06:36:33Z <p>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 <em>not</em> 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. </p> <p>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.</p> <blockquote> <p>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 <strong>general</strong> 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 $d$. </p> <p>Is it true that $Q$ itself is contained in a plane (and is the complete intersection of two curves of degree $d$)?</p> <p>If not, what is a counterexample?</p> </blockquote> <p>Any answer or reference to the existing literature will be appreciated. Thank you. </p> <p><strong>EDIT.</strong> 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 $d$)? </p> http://mathoverflow.net/questions/67265/when-is-a-general-projection-of-d2-points-in-mathbbp3-a-complete-interse/67278#67278 Answer by Dmitri for When is a general projection of $d^2$ points in $\mathbb{P}^3$ a complete intersection? Dmitri 2011-06-08T16:14:11Z 2011-06-08T23:20:43Z <p><strong>Counterexample.</strong> 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 $L_i\cap M_j$. </p> <p><strong>Proof.</strong> 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. </p> <p>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? </p>