Complete intersections in relation to a paper of B. J. Birch (Q2759196)

Let \(f_j(x)\), \(1\leq j\leq l\), be a non-constant homogeneous polynomial in \(n\) variables \(x:= (x_1,\dots, x_n)\) with coefficients in a perfect field \(k\). Let us consider an affine algebraic set \(V_\mu: f_1(x)= \mu_1,\dots, f_l(x)= \mu_l\), let \(V_\mu^*\) stand for the singular locus of \(V_\mu\), and let \(V^*\) denote the closure of the union of \(V_\mu^*\), as \(\mu:= (\mu_1,\dots, \mu_l)\) ranges over \(k^l\). Further, let \(J_l\) denote the ideal in \(k[ x]\) generated by the minors of order \(l\) of the Jacobian matrix \((\frac {\partial f_j}{\partial x_i})\), and let \(W= \text{Spec\,} k[ x]/J_l\). Assuming \(\dim V_0= n-l\), we prove that the affine algebraic set \(V^*\) is homeomorphic (but not necessarily isomorphic!) to \(W\) and that \(V^*(k)= W(k)\). This justifies an assertion used in a well-known paper of \textit{B. J. Birch} [Proc. R. Soc. Lond., Ser. A 265, No. 1321, 245--263 (1962; Zbl 0103.03102)]. The cited result is deduced from a new property of complete intersections in a graded Noetherian ring proved in this paper, in conjunction with the Jacobian criterion and the Macaulay unmixedness theorem.