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Prescribed intersection of varieties

Every variety here is complex analytic, or complex algebraic if it solves anything. Given a germ of a (possibly singular, nor necessarily irreducible) hypersurface $(H,0)\subset(\mathbb{C}^{n+1},0)$ ...
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How to prove that a specific quadric intersection is complete and irreducible?

Let's borrow the quadric intersection $I$ from another question. More precisely, let $k$ be an algebraically closed field of characteristic $\neq 2$ and $a_1, a_2, \cdots, a_n \in k^*$ be some ...
Dimitri Koshelev's user avatar