is irreducible and very free.

We need to analyze the normal bundle of the minimal very free curve. The following construction, used in [@ShenMM], is very helpful.

Let $C$ be a very free curve with $-K_X$ degree equal to either $7$ or $8$ and general in an irreducible component of moduli space of very free curve. Assume the normal bundle of $C$ is ${{\mathcal}{O}}(1)\oplus {{\mathcal}{O}}(1)\oplus{{\mathcal}{O}}(a), a=3 ~\text{or}~4$.

Choose $2$ points in $C$ and deform $C$ with the two points fixed. Then the deformation of $C$ sweeps out a surface $\Sigma$ in $Y$. Let $\Sigma'$ be the normalization and $\tilde{\Sigma}$ be the minimal resolution of $\Sigma'$.

We need the following definition.

Let $C_i \subset X_i$ be a curve on a variety $X_i$, $i=1, 2$. We say $(X_1,C_1)$ is *equivalent to* $(X_2, C_2)$ if there is an open neighborhood $V_i$ of $C_i$ in $X_i$ and an isomorphism $f: V_1 \rightarrow V_2$ such that $f\vert_{C_1}:C_1 \rightarrow C_2$ is also an isomorphism.

The following results are proved in [@ShenMM], Section 2.2, 2.3.

\[prop:Shen\] Notations as above.

1.  $\Sigma$ is independent of the choice of the points. ${\Sigma}'$ is smooth along $C$ and $N_{C/\Sigma'}\cong {{\mathcal}{O}}(a)$.

2.  There is a neighborhood $U$ of $C$ such that 