Lemma 93.17.2. Let k be a field. Let 1 \leq c \leq n be integers. Let f_1, \ldots , f_ c \in k[x_1, \ldots x_ n] be elements. Let a_{ij}, 0 \leq i \leq n, 1 \leq j \leq c be variables. Consider
g_ j = f_ j + a_{0j} + a_{1j}x_1 + \ldots + a_{nj}x_ n \in k[a_{ij}][x_1, \ldots , x_ n]
Denote Y \subset \mathbf{A}^{n + c(n + 1)}_ k the closed subscheme cut out by g_1, \ldots , g_ c. Denote \pi : Y \to \mathbf{A}^{c(n + 1)}_ k the projection onto the affine space with variables a_{ij}. Then there is a nonempty Zariski open of \mathbf{A}^{c(n + 1)}_ k over which \pi is smooth.
Proof.
Recall that the set of points where \pi is smooth is open. Thus the complement, i.e., the singular locus, is closed. By Chevalley's theorem (in the form of Morphisms, Lemma 29.22.2) the image of the singular locus is constructible. Hence if the generic point of \mathbf{A}^{c(n + 1)}_ k is not in the image of the singular locus, then the lemma follows (by Topology, Lemma 5.15.15 for example). Thus we have to show there is no point y \in Y where \pi is not smooth mapping to the generic point of \mathbf{A}^{c(n + 1)}_ k. Consider the matrix of partial derivatives
(\frac{\partial g_ j}{\partial x_ i}) = (\frac{\partial f_ j}{\partial x_ i} + a_{ij})
The image of this matrix in \kappa (y) must have rank < c since otherwise \pi would be smooth at y, see discussion in Smoothing Ring Maps, Section 16.2. Thus we can find \lambda _1, \ldots , \lambda _ c \in \kappa (y) not all zero such that the vector (\lambda _1, \ldots , \lambda _ c) is in the kernel of this matrix. After renumbering we may assume \lambda _1 \not= 0. Dividing by \lambda _1 we may assume our vector has the form (1, \lambda _2, \ldots , \lambda _ c). Then we obtain
a_{i1} = - \frac{\partial f_ j}{\partial x_1} - \sum \nolimits _{j = 2, \ldots , c} \lambda _ j(\frac{\partial f_ j}{\partial x_ i} + a_{ij})
in \kappa (y) for i = 1, \ldots , n. Moreover, since y \in Y we also have
a_{0j} = -f_ j - a_{1j}x_1 - \ldots - a_{nj}x_ n
in \kappa (y). This means that the subfield of \kappa (y) generated by a_{ij} is contained in the subfield of \kappa (y) generated by the images of x_1, \ldots , x_ n, \lambda _2, \ldots , \lambda _ c, and a_{ij} except for a_{i1} and a_{0j}. We count and we see that the transcendence degree of this is at most c(n + 1) - 1. Hence y cannot map to the generic point as desired.
\square
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