Tropical floor plans and enumeration of complex and real multi-nodal surfaces

The family of complex projective surfaces in $\mathbb {P}^3$ of degree $d$ having precisely $\delta$ nodes as their only singularities has codimension $\delta$ in the linear system $|{\mathcal O}_{\mathbb {P}^3}(d)|$ for sufficiently large $d$ and is of degree $N_{\delta ,\mathbb {C}}^{\mathbb {P}^3}(d)=(4(d-1)^3)^\delta /\delta !+O(d^{3\delta -3})$. In particular, $N_{\delta ,\mathbb {C}}^{\mathbb {P}^3}(d)$ is polynomial in $d$.

By means of tropical geometry, we explicitly describe $(4d^3)^\delta /\delta !+O(d^{3\delta -1})$ surfaces passing through a suitable generic configuration of $n=\binom {d+3}{3}-\delta -1$ points in $\mathbb {P}^3$. These surfaces are close to tropical limits which we characterize combinatorially, introducing the concept of floor plans for multinodal tropical surfaces. The concept of floor plans is similar to the well-known floor diagrams (a combinatorial tool for tropical curve counts): with it, we keep the combinatorial essentials of a multinodal tropical surface $S$ which are sufficient to reconstruct $S$.

In the real case, we estimate the range for possible numbers of real multi-nodal surfaces satisfying point conditions. We show that, for a special configuration $\boldsymbol {w}$ of real points, the number $N_{\delta ,\mathbb {R}}^{\mathbb {P}^3}(d,\boldsymbol {w})$ of real surfaces of degree $d$ having $\delta$ real nodes and passing through $\boldsymbol {w}$ is bounded from below by $\left (\frac {3}{2}d^3\right )^\delta /\delta ! +O(d^{3\delta -1})$.

We prove analogous statements for counts of multinodal surfaces in $\mathbb {P}^1\times \mathbb {P}^2$ and $\mathbb {P}^1\times \mathbb {P}^1\times \mathbb {P}^1$.