Strong and weak conjectures on essential specializations and the discriminant ideal

Both conjectures are false, even when the singular-specialization locus is nonempty and both discriminant ideals are proper.

The construction amplifies the one-parameter example in M. Wibmer, Gröbner bases for families of affine or projective schemes, Journal of Symbolic Computation 42 (2007), Example 3, https://arxiv.org/abs/math/0608019. A related example was also recorded by A. Montes, On the canonical discussion of polynomial systems with parameters, https://arxiv.org/abs/math/0601674.

Let

$$R=\mathbb Q[u,v]$$

and consider

$$I=\langle vu(ux-1),\,vx(ux-1)\rangle =v(ux-1)\langle u,x\rangle \subseteq R[x].$$

Over $\mathbb Q(u,v)$, the generic reduced Gröbner basis is

$$G=\{g\},\qquad g=x-\frac1u,$$

with leading-power-product set $\{x\}$.

For $(u,v)=(c,b)$, the specialized ideal is

$$I_{c,b}= \begin{cases} \langle x-1/c\rangle,&b\neq0,\ c\neq0,\\ \langle x\rangle,&b\neq0,\ c=0,\\ (0),&b=0. \end{cases}$$

Under the original definition, a specialization is singular precisely when its leading-power-product set differs from the generic set. Therefore the singular locus is exactly $V(v)$, and hence

$$N=I(V(v))=(v). \tag{1}$$

On the other hand,

$$J_g=\{a\in R:ag\in I\}.$$

If $ag\in R[x]$, its constant coefficient $-a/u$ forces $a=uh$ for some $h\in R$. Cancelling $ux-1$ in the integral domain $R[x]$ gives

$$ag=h(ux-1)\in v(ux-1)(u,x) \quad\Longleftrightarrow\quad h\in v(u,x)\cap R=(uv).$$

Thus

$$J_g=(u^2v), \qquad J=\sqrt{J_g}=(uv). \tag{2}$$

At the point $(u,v)=(0,1)$,

$$J_g\subseteq(u,v-1),$$

so the specialization is essential. Nevertheless its specialized ideal is $\langle x\rangle$, whose leading-power-product set is still $\{x\}$; therefore it is not singular. This disproves the strong conjecture.

Finally, (1) and (2) give

$$J=(uv)\subsetneq(v)=N,$$

so $v\in N\setminus J$. Hence the conjectured inclusion $J\supseteq N$ also fails, disproving the weak conjecture.