Let

$A \colon= {\Bbb C}[S_1,\ldots,S_n]$ with $1 \leq n < \infty$

$B \colon= A[X_1,\ldots,X_d]$ with $2 \leq d < \infty$.

$O \colon= (0,\ldots,0)$ be the origin of ${\mathrm{Spec}}\,B$.

Suppose ${\frak P}$ be a prime ideal of $B$ such that ${\frak P} \cap A = 0$.

Scheme-theoretically, this is equivalent to the following condition$\colon$ ${\mathrm{Spec}}\,B/{\frak P} \to {\mathrm{Spec}}\,A$ is dominant, i.e. surjective after taking the closure of the image.

Let $H$ be a hyperplane of ${\mathrm{Spec}}\,B$ whose defining equation $h$ satisfies the following condition$\colon$ $h = \Sigma_{i=1}^{i=d}a_iX_i\, ;\, a_i \in B, \phantom{i}^{\exists}a_i \not= 0 \,\,{\mathrm{s.t.}}\,\, a_i \in {\Bbb C}$.

and define the intersection $D \colon= ({\mathrm{Spec}}\,B/{\frak P}) \cap H$. There is a natural morphism $\phi \colon D \to {\mathrm{Spec}}\,A$.

## Q. Is it possible to choose $H$ such that the following three conditions are satisfied?

(i) $D$ passes through $O$.

(ii) $D$ is irreducible.

(iii) $\phi$ is dominant.