k is a perfect field. X and Y are two regular varieties over k. Does their fiber product over k remain to be regular?
Note: When k is algebraically closed it's true by Jacobian criterion. When k is not perfect there's counterexample.
k is a perfect field. X and Y are two regular varieties over k. Does their fiber product over k remain to be regular? Note: When k is algebraically closed it's true by Jacobian criterion. When k is not perfect there's counterexample. 


The answer is yes. Indeed, over a perfect field the notions of smooth and regular coincide so it follows from the fact that base change and composition preserve smoothness. 

