Towards tropically counting binodal surfaces

TL;DR

This paper advances tropical enumerative geometry by counting multinodal surfaces with close nodes using floor plans, revealing their asymptotic contribution to complex surface degrees and classifying tropical node configurations.

Contribution

It generalizes tropical floor plans to count multinodal curves and classifies when unseparated nodes tropicalize to specific vertices, linking tropical and complex surface counts.

Findings

Tropical multinodal surface counts are obtained using floor plans.

Unseparated nodes contribute to the second order term in degree polynomials.

Classification of tropical node configurations for degree d > 4 surfaces.

Abstract

Tropical counting tools are useful for many enumerative questions. We count tropical multinodal surfaces using floor plans, looking at the case when two nodes are tropically close together, i.e., unseparated. We generalize tropical floor plans to recover the count of multinodal curves. We then prove that for $\delta=2$ or $3$ nodes, tropical surfaces with unseparated nodes contribute asymptotically to the second order term of the polynomial giving the degree of the family of complex projective surfaces in $\mathbb{P}^3$ of degree $d$ with $\delta$ nodes. We classify when two nodes in a surface tropicalize to a vertex dual to a polytope with 6 lattice points, and prove that this only happens for projective degree $d$ surfaces satisfying point conditions in Mikhalkin position when $d>4$.