Tropical floor plans and enumeration of complex and real multi-nodal surfaces
Hannah Markwig, Thomas Markwig, Kristin Shaw, Eugeniĭ Shustin · Journal of Algebraic Geometry · 2021
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