# Morphisms of smooth varieties

Let $$f:X\rightarrow Y$$ be a surjective morphism of varieties (integral separated schemes of finite type over an algebraically closed field) such that $$Y$$ is smooth projective and all the fibers are connected smooth projective. Is $$X$$ necessarily a smooth variety? If $$f$$ were flat, smoothness of $$X$$ would be automatic but we consider non-flat morphisms as well.

• Welcome new contributor. That is not true. Let $Y$ be $\mathbb{P}^2_k = \text{Proj}\ k[s,t,u]$. Let $Z$ be $\mathbb{P}^1_k = \text{Proj}\ k[x,y]$. Inside the product scheme $Y\times_{\text{Spec}\ k} Z$, let $X$ be the zero scheme of the following bihomogeneous polynomial of bidegree $(2,1)$: $p = s^2x+t^2y$. Let $f$ be the restriction to $X$ of the projection to $Y$. This is probably the simplest example of a projective Abelian cone that is irreducible yet not smooth. To learn more about this, confer Section 2 of the following: arxiv.org/pdf/math/0305432.pdf – Jason Starr Jul 12 at 14:25
• Welcome new contributor. It appears that there may be some issues with your use of this site. If you have similar questions, please feel free to e-mail me directly. – Jason Starr Jul 12 at 14:27