Let $\mathbb K$ be an algebraically closed field (of characteristic zero) and $H$ an irreducible variety in $\mathbb K ^n$. Let $t \in \mathbb K [x_1,\ldots,x_n]$ and let $T:= \mathsf V ( t )$ be the variety defined by $t$ and let $nT:=\mathsf V(t\cdot z - 1)$ where $z$ is an new variable. Let $r$ be an integer smaller than $n$.
Assuming that the projection of $H$ to $\mathbb K ^r$ is the whole space $\mathbb K ^r$.
Can it then happen that the projection of $H \cap T$ to $\mathbb K ^r$ and the projection of $H \cap nT$ to $\mathbb K ^r$ are both $\mathbb K ^r$?
My guess is no but I can't prove it.
This question is about algebraic geometry proving. If $H$ is the variety of the hypotheses and $t$ is the thesis, $n$ is the number of variables and $r$ the number of free variables. If $H$ is irreducible, can it then happen that the theorem is neither generically true (the projection of $H \cap nT$ to $\mathbb K ^r$ is not $\mathbb K ^r$) nor generically false (the projection of $H \cap T$ to $\mathbb K ^r$ is not $\mathbb K ^r$)?
Thank you in advance!