Enumeration of Complex and Real Surfaces via Tropical Geometry

TL;DR

This paper establishes a correspondence between singular tropical and algebraic surfaces in three dimensions, introduces a 3D lattice path algorithm for enumeration, and demonstrates the abundance of real singular surfaces in certain pencils.

Contribution

It develops a 3D tropical enumeration method for singular surfaces and proves the existence of many real singular surfaces, extending previous 2D results.

Findings

A correspondence theorem linking tropical and algebraic singular surfaces.

A 3D lattice path algorithm for enumerating singular tropical surfaces.

Existence of pencils with at least (3/2)d^3 singular real surfaces.

Abstract

We prove a correspondence theorem for singular tropical surfaces in real three space, which recovers singular algebraic surfaces in an appropriate toric three-fold that tropicalize to a given singular tropical surface. Furthermore, we develop a three-dimensional version of Mikhalkin's lattice path algorithm that enumerates singular tropical surfaces passing through an appropriate configuration of points in real three space. As application we show that there are pencils of real surfaces of degree $d$ in projective three space containing at least $(3/2)d^3+O(d^2)$ singular surfaces, which is asymptotically comparable to the number $4(d-1)^3$ of all complex singular surfaces in the pencil. Our result relies on the classification of singular tropical surfaces (arXiv:1106.2676).