# 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?