Let $K$ be a of characteristic 0. A cubic surface in $\mathbb{P}^3$ is the zero set of a homogeneous polynomial $f = c_1 x^3 + \cdots + c_{20} yzw$, where $c_i \in K$. The coefficients of this polynomial give a $\mathbb{P}^{19}$ of cubic surfaces. For any point $a \in \mathbb{P}^3$, the collection of all cubics through $a$ is a hyperplane in $\mathbb{P}^{19}$.

A node $c \in \mathbb{P}^3$ is a point where all of the partial derivatives of $f$ simultaneously vanish but the Hessian matrix of second partial derivatives is not singular. Asking that a surface contains a node inflicts a codimension 1 condition on the $\mathbb{P}^{19}$ of all cubics. Let $k$ be $2,3,4$. The variety $X_k$ of $k$-nodal cubics is irreducible of codimension $k$ in $\mathbb{P}^{19}$. The respective degrees of these varieties are 280, 800, and 305. Since the degree is the number of points contained in the intersection of the variety $X_k$ with a general linear space of dimension equal to the codimension of $X_k$, this number also counts the number of $k$-nodal cubics through codimension $X_k$ many general points.


Can the numbers 280, 800, 305 of $k$-nodal cubics through $19-k$ general points for $k = 2,3,4$ be derived tropically?

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