Factorization of quartic forms

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Consider (complex) quartic forms of three variables that can be factored as product of quadratic forms. Being the image of a regular map from $\mathbb{P}(S_2) \times \mathbb{P}(S_2)$ to $\mathbb{P}(S_4)$, it is a projective variety. Do we know explicitly polynomials cut it set-theoretically?