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

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

By means of tropical geometry, we explicitly describe ( 4 d 3 ) δ / δ ! + O ( d 3 δ − 1 ) (4d^3)^\delta /\delta !+O(d^{3\delta -1}) surfaces passing through a suitable generic configuration of n = ( d + 3 3 ) − δ − 1 n=\binom {d+3}{3}-\delta -1 points in P 3 \mathbb {P}^3