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I think this is false, because of the following example.

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 $g \geq 2$, since 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].

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].

I think this is false, because of the following example.

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 dimension of the tangent space at the Hilbert scheme $\mathscr{H}$ of $\Delta$ in $C \times C$ at the point $[\Delta]$ corresponding to $\Delta$ is 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 $g \geq 2$, since in that case $\Delta^2 < 0$.

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

I think this is false, because of the following example.

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 $g \geq 2$, since 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].

I think this is false, because of the following example.

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 $\Delta^2= 2-2g$, so it is rigid as soon as $g \geq 2$.

I think this is false, because of the following example.

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 dimension of the tangent space at the Hilbert scheme $\mathscr{H}$ of $\Delta$ in $C \times C$ at the point $[\Delta]$ corresponding to $\Delta$ is 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 $g \geq 2$, since in that case $\Delta^2 < 0$.

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