Question

Let $X=Y\times Z$ be a product of complex manifolds $Y,Z$. Is it true that there exists no rigid curve on $X$? Here I mean by a rigid curve a curve which is not a member of any family of curves on $X$.

Answer 1

These curves may actually exist, as the following example shows.

Let $C$ be a smooth curve of genus $g$. Then the diagonal $\Delta \subset C \times C$ is isomorphic to $C$ and has self intersection $\Delta^2= 2-2g$.

On the other hand, the tangent space of the Hilbert scheme $\mathscr{H}$ of $\Delta$ in $C \times C$ at the point $[\Delta]$ has dimension given by $$ \dim _{[\Delta]} \mathscr{H} = h^0(\Delta, \, N_{\Delta/ C \times C}) = h^0(\Delta, \mathscr{O}_{\Delta}(\Delta))$$
and this quantity is $0$ for $g \geq 2$, because in that case $\Delta^2 < 0$.

This precisely means that $\Delta$ is rigid in $C \times C$ as soon as $g \geq 2$.

The cone of effective curves in $C \times C$ ($g \geq 2$) is quite subtle, and not fully understood in general. For further details, see [R. Lazarsfeld, Positivity in Algebraic Geometry I, Section 1.5].