Let $S$ be a smooth projective surface (I am mostly intrested in the case when $S$ is a product of curves, say $S=\mathbb{P}^1 \times \mathbb{P}^1$ but probably this is not important).

Consider a family of curves $X \subset S \times T$ parametrised by a variety $T$ of dimension 2 (the fibres $X_t$ are distinct). Denote the projection $X \to S\ $ as $p_S$ and $X \to T\ $ as $p_T$. Define a map $\tau: X \to \mathbb{P}TS$ which to every point $x \in X$ associates the projectivisation of the tangent vector of $X_{p_T(x)}$ at $p_S(x)$.

My question is: how can one show that there exists $s \in S$ such that the map $\tau$ is non-constant on $p^{-1}(s)$?

In other words, how can one show that there is a point $s \in S$ such that not all curves passing through it touch each other, but on the contrary, their tangent spaces sweep the tangent space of $S$ at this point?

**update**: Consider the map $\sigma: \mathbb{P}(TX/T) \to \mathbb{P}(TS)$ induced by the projection $p_S: X \to S$. If $\sigma^{-1}(v)$ were finite for some $v \in TS$, then the projection of $v$ to $S$ would be the answer to the question. If the image of the map $\sigma$ is of dimension 3 then almost all fibres are of dimension 0, since $\mathbb{P}(TX/T)$ is of dimension 3. Perhaps one can show that $\mathrm{dim}\ \mathrm{Im}\ \sigma=3$?