\tilde{C}}$.

We choose the constraints to be three points. And in view of Lemma \[lem:smoothing\] and \[lem:smoothable\], we only need to consider the following possibilities.

1.  There are four irreducible components $C_1$ through $C_4$ with $-K_X$ degree $2, 2, 2$, and $2$. The curves $C_1, C_2$, and $C_3$ are free and pass through one of the chosen general points.

2.  There are four irreducible components $C_1$ through $C_4$ with $-K_X$ degree $2, 2, 3$, and $1$. The curves $C_1, C_2$, and $C_3$ are free and pass through one of the chosen general points.

3.  There are five irreducible components $C_1$ through $C_5$ with $-K_X$ degree $2, 2, 2, 1, 1$. The curves $C_1, C_2$, and $C_3$ are free and pass through one of the chosen general points.

In the first case, if the Kontsevich moduli space has the expected dimension at the curve $C_4$, then the union of $C_1, \ldots, C_4$ can be smoothed to a free curve by Lemma \[lem:smoothable\]. Thus it lies in an irreducible component of expected dimension. So this will not happen if we choose the $3$ points to be general. So the dimension of the Kontsevich moduli space at the point $[C_4]$ has strictly higher 