Let $f:X \rightarrow T$ be a proper morphism from a projective scheme $X$ to a smooth projective curve $T$ over $\mathbb{C}$. I know that fiber dimension is upper-semicontinuous, but is the degree of fibers also upper-semicontinuous? Here I fix the embedding of $X$ into some projective space (and so all the fibers of $f$ over each closed point of $T$) and then define the degree inside this fixed projective space in an usual way.

Actually, what I have in mind is the following situation: Let $\mathbb{P}^N$ be a projective space over $\mathbb{C}$ and let $Z$ and $Z_t$ be two irreducible varieties, where $Z$ is fixed and $Z_t$ moves with $t \in T$ as a parameter on a smooth curve $T$. I want to prove upper semicontinuity of degrees of the intersection scheme of $Z$ and $Z_t$ in $\mathbb{P}^N$, when dimension of $Z$ and $Z_t$ are smaller than $N \over 2$ but they intersect. In other words, the intersection is not proper in $\mathbb{P}^N$ and the expected dimension of the intersection is negative. For example, you could consider a line L in $\mathbb{P}^N$ and a family of curves $C_t$ with t as its parameter such that $L \cap C_t$ is nonempty, when $N>2$.