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

The family of complex projective surfaces in P3\mathbb {P}^3 of degree dd having precisely δ\delta nodes as their only singularities has codimension δ\delta in the linear system |OP3(d)||{\mathcal O}_{\mathbb {P}^3}(d)| for sufficiently large dd and is of degree Nδ,CP3(d)=(4(d−1)3)δ/δ!+O(d3δ−3)N_{\delta ,\mathbb {C}}^{\mathbb {P}^3}(d)=(4(d-1)^3)^\delta /\delta !+O(d^{3\delta -3}). In particular, Nδ,CP3(d)N_{\delta ,\mathbb {C}}^{\mathbb {P}^3}(d) is polynomial in dd. By means of tropical geometry, we explicitly describe (4d3)δ/δ!+O(d3δ−1)(4d^3)^\delta /\delta !+O(d^{3\delta -1}) surfaces passing through a suitable generic configuration of n=(d+33)−δ−1n=\binom {d+3}{3}-\delta -1 points in P3\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 SS which are sufficient to reconstruct SS. 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 w\boldsymbol {w} of real points, the number Nδ,RP3(d,w)N_{\delta ,\mathbb {R}}^{\mathbb {P}^3}(d,\boldsymbol {w}) of real surfaces of degree dd having δ\delta real nodes and passing through w\boldsymbol {w} is bounded from below by (32d3)δ/δ!+O(d3δ−1)\left (\frac {3}{2}d^3\right )^\delta /\delta ! +O(d^{3\delta -1}). We prove analogous statements for counts of multinodal surfaces in P1×P2\mathbb {P}^1\times \mathbb {P}^2 and P1×P1×P1\mathbb {P}^1\times \mathbb {P}^1\times \mathbb {P}^1.