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

TL;DR

This paper uses tropical geometry to explicitly count and describe complex and real multinodal surfaces in projective three space, introducing floor plans to understand their combinatorial structure and estimating the number of real solutions.

Contribution

It introduces the concept of floor plans for multinodal tropical surfaces and provides explicit counts and bounds for complex and real multinodal surfaces.

Findings

Explicit enumeration formulas for complex multinodal surfaces.

Lower bounds for the number of real multinodal surfaces.

Extension of methods to other projective varieties.

Abstract

The family of complex projective surfaces in projective three space of degree $d$ having precisely $\delta$ nodes as their only singularities has codimension $\delta$ in the linear system of surfaces of degree $d$ for sufficiently large $d$ and is of degree $N_{\delta,complex}(d)=(4(d-1)^3)^\delta/\delta!+O(d^{3\delta-3})$. In particular, this number 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 projective three space. 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 which are sufficient to reconstruct the surface. 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 $w$ of real points, the number $N_{\delta,real}(d,w)$ of real surfaces of degree $d$ having $\delta$ real nodes and passing through $w$ is bounded from below by $(\frac{3}{2}d^3)^\delta/\delta! +O(d^{3\delta-1})$. We prove analogous statements for counts of multinodal surfaces in $P^1\times P^2$ and $P^1\times P^1\times P^1$.