Equations for the locus of k-nodal cubics

Setup and question

For the values $k=1,2,3,4$, the variety of k-nodal cubic surfaces is irreducible of codimension $k$ in $\mathbb{P}^{19}$. What is the degree of these varieties? (The degree question was answered in greater generality by Vainsencher in 2003, see below.) Can we find explicit low-degree polynomials that vanish on these varieties?

Comments

Here is a list of motivating references as the point of departure:

Rennemo proved that the number of hypersurfaces with a certain type of singularities in an arbitrary linear system is always governed by a polynomial in Chern classes of the associated line bundle. Relevant to the existence of the polynomial is the recent work of Tzeng.

Vainsencher described these polynomials explicitly in a $k$-dimensional family of hypersurfaces with $k \leq 6$ ordinary double points. Evaluating his formulas in the particular case of cubic surfaces for $k=1,2,3,4$, one obtains that the reduced degrees of the varieties of the $k$-nodal cubics are $32,280,800,305$, respectively.

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