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Suppose that I am given a subscheme $Y$ of $\mathbf{P}^n_{\mathbf{Z}}$, flat over $\operatorname{Spec}\mathbf{Z}$ and with smooth generic fiber $Y_{\mathbf{Q}}$, defined by the vanishing of some homogeneous polynomials $$F_1, \ldots, F_k \in \mathbf{Z}[X_0,\ldots,X_n].$$ How does one determine the set $S$ consisting of primes $p$ such that the fiber $Y_{\mathbf{F}_p}$ is non-smooth?
For simplicity suppose that $k=1$, and write $F=F_1$. In a nutshell, I compute the derivatives $f_i := \frac{\partial F}{\partial X_i}$, and determine an integer $N$ such that $N$ is contained in the ideal $I = (F, f_1, \ldots, f_n) \subset \mathbf{Z}[X_0,\ldots,X_n]$ by repeatedly taking resultants of polynomials in $I$. Then at least for all $p$ not dividing $N$, the fiber $Y_{\mathbf{F}_p}$ is smooth, since for such $p$ the ideal $I_p \subset \mathbf{F}_p[X_0,\ldots,X_n]$ generated by the reduced polynomials $\widetilde{F}$, $\widetilde{f}_1$,$\ldots$,$\widetilde{f}_n$ contains $1$.