Topological complexity of enumerative problems and classifying spaces of $PU_n$

TL;DR

This paper investigates the topological complexity of classical enumerative geometry problems, providing lower bounds and analyzing classifying spaces of projective unitary groups to understand the computational difficulty involved.

Contribution

It introduces new lower bounds for the topological complexity of solving key enumerative problems and explores the cohomology of classifying spaces of $PU_n$ groups.

Findings

Established lower bounds for topological complexity of three classical problems.

Analyzed cohomology classes of classifying spaces of $PU_n$.

Connected topological complexity with algebraic geometry enumerative problems.

Abstract

We study the topological complexity, in the sense of Smale, of three enumerative problems in algebraic geometry: finding the 27 lines on cubic surfaces, the 28 bitangents and the 24 inflection points on quartic curves. In particular, we prove lower bounds for the topological complexity of any algorithm that finds solutions to the three problems and for the Schwarz genera of their associated covers. The key is to understand cohomology classes of the classifying spaces of projective unitary groups $PU_n$.