Suppose I have an irreducible affine variety $X = \{ \mathbf{x} \in \mathbb{A}^n_{\mathbb{C}}: f_i(\mathbf{x}) = 0, 1 \leq i \leq m \}$ for some integer $m$. Let us define $X_j = \{ \mathbf{x} \in \mathbb{A}^n_{\mathbb{C}}: f_i(\mathbf{x}) = 0, 1 \leq i \leq m, i \neq j \}$. Is it always the case that $X_j$ is also irreducible? Would this possibly be the case if I assume each $f_i$ is homogeneous? I'm not quite sure where to get started... Suggestions are appreciated!
2026-03-27 16:53:53.1774630433
removing one polynomial equation from irreducible affine variety keeps irreducibility?
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No. For instance $$ x_1x_3 - x_2^2 = x_1x_4 - x_2x_3 = x_2x_3 - x_3^2 = 0 $$ defines in $\mathbb{A}^4$ the cone over a twisted cubic curve, which is irreducible, but if you drop any of the equations, the resulting variety is the union of that cone with a hyperplane.