10

# Setup

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.

# Question

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

# Partial Progress

Madeline Brandt and Alheydis Geiger studied the case when the two tropical nodes are far enough apart to behave like single nodes. They count 214 binodal tropical cubic surfaces with separated nodes through points in Mikhalkin position.
https://arxiv.org/abs/1909.09105