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

Weiyan Chen, Xing Gu

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 $PU_n$.

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

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 .

Paper Structure

This paper contains 8 sections, 25 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. General results about Schwarz genus
  4. On the cohomology of $B\mathrm{PU}_n$
  5. Cohomology of $\mathrm{PU}_n$ homogeneous spaces
  6. The 27 lines on cubic surfaces
  7. The 24 inflection points and 28 bitangents on quartic curves
  8. Proof of Theorem \ref{['TC main theorems']}

Key Result

Theorem 1.1

When $\epsilon$ is sufficiently small, we have

Theorems & Definitions (37)

  • Theorem 1.1
  • Definition 1: Schwarz
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1: Schwarz, p.71
  • Proposition 2.2: Schwarz, p.71
  • Corollary 2.3: Schwarz, p.76
  • proof : Proof (Proposition \ref{['pro:fiberwise join']} $\Rightarrow$ Corollary \ref{['cor:coho dim']})
  • Proposition 2.4: Schwarz, p.98
  • ...and 27 more