Multiplicity of the intersection of a Rational curve in a quadric with a tangent plane

Question

Consider a rational map $u : \mathbb{CP}^1 \to \mathbb{CP}^4$ of degree~$d$, such that the image lies in a fixed 3-dimensional quadric $Q^3$. In other words, its image is a rational curve in $Q^3 \subset \mathbb{CP}^4$. Now consider a point $p$ on this curve and look at a linear hypersurface $T_p \subset \mathbb{CP}^4$ which is tangent to the quadric $Q$ at $p$. It is also tangent to the curve defined by the parametrization $u$ and therefore the rational curve intersects $T_p$ at $p$ with multiplicity at least 2. If the multiplicity is more than 2, is there an arbitrary small perturbation of this curve to a rational curve of the same degree inside the quadric that will intersect $T_p$ with multilicity exactly 2?

Answer 1

If I understand the question correctly, the answer is yes (if $d$ is at least $2$), by irreducibility of the stable maps space + semicontinuity + existence of a nodal rational curve satisfying what you want. For the nodal rational curve, just take a conic through $p$ and some lines intersecting the conic at a single point away from $p$.

I don't know what the best approach to irreducibility of the moduli space of genus $0$ stable maps to $Q^3$ is. It's equivalent to the irreducibility of the moduli space of genus $0$ stable maps to $\mathbb{CP}^3$, which is well-known, but I don't know a reference. Alternatively, you can just cite "Rational curves on hypersurfaces of low degree", Harris-Roth-Starr, which proves something significantly more general.