ag.algebraic geometry – Negative curves on surfaces

Let $f:Xrightarrowmathbb{P}^1$ be a family of surfaces over $mathbb{C}$. Assume that for $yinmathbb{P}^1$ general $X_y = f^{-1}(y)$ is a smooth surface. Let $X_{eta}$ be the generic fiber of $f$ and $overline{X}_{eta} := X_{eta}times_{Spec(mathbb{C}(t))}Spec(overline{mathbb{C}(t)})$, where $overline{mathbb{C}(t)}$ is the algebraic closure of $mathbb{C}(t)$.

Assume that in $overline{X}_{eta}$ there is a curve of negative self-intersection $-a$. Does there is a curve of negative self-intersection $-a$ on the fiber $X_y = f^{-1}(y)$ for $yinmathbb{P}^1$ general?