Let $p\_1,...,p\_8\in\mathbb{P}^{3}$ be points in linear general position. Then there exists a unique elliptic curve $C$ of degree $4$ passing through $p\_1,...,p\_8$. I am interested in what happens for nodal rational curves of degree $4$.


Now, let us suppose we have seven points $p\_1,...,p\_7\in\mathbb{P}^{3}$ in linear general position. 


**How many irreducible, rational curves of degree $4$ pass through $p\_2,...,p\_7$ and have a singular point of multiplicity $2$ at $p\_1$ ?** 


Such a curve can be constructed as a complete intersection of two quadric surfaces which are tangent at $p\_1$, or as a projection of a rational normal quartic curve $C$ in $\mathbb{P}^{4}$ from a point lying on a secant line or an a tangent line of $C$.



Counting curves of degree 4 in $\mathbb{P}^{3}$
Zero. Indeed, if the intersection $Q\_1 \cap Q\_2$ of two quadrics is singular at $p\_1$, then there is a quadric $Q$ in the pencil generated by $Q\_1$ and $Q\_2$ which is singular at $p\_1$. On the other hand, if a quadric cone with vertex at $p\_1$ passes through $p\_2,\dots,p\_7$, then the images of $p\_2,\dots,p\_7$ in $\mathbb P^2$, obtained by the projection from $p\_1$, all lie on a conic. But a general 6-tuple of points in $\mathbb P^2$ does not lie on a conic. 


