Contracted divisors and Degree-Two Maps

We consider polynomial maps of affine space over an algebraically closed field of characteristic zero. We prove that every irreducible component of the zero locus of the Jacobian determinant corresponds to either a contracted divisor or a branching divisor. We further consider polynomial maps of degree two without contracted divisors and show that the Jacobian determinant is irreducible, anti-invariant under the Galois involution, and coincides with the defining equation of the unique branching divisor.