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Flops and Hilbert schemes of space curve singularities

Duiliu-Emanuel Diaconescu, Mauro Porta, Francesco Sala, Arian Vosoughinia

Abstract

Using pagoda flop transitions between smooth projective threefolds, a relation is derived between the Euler numbers of moduli spaces of stable pairs which are scheme-theoretically supported on a fixed singular space curve and Euler numbers of Flag Hilbert schemes associated to a plane curve singularity. When the space curve singularity is locally complete intersection, one obtains a relation between the latter and Euler numbers of Hilbert schemes of the space curve singularity. It is also shown that this relation yields explicit results for a class of torus-invariant locally complete intersection singularities.

Flops and Hilbert schemes of space curve singularities

Abstract

Using pagoda flop transitions between smooth projective threefolds, a relation is derived between the Euler numbers of moduli spaces of stable pairs which are scheme-theoretically supported on a fixed singular space curve and Euler numbers of Flag Hilbert schemes associated to a plane curve singularity. When the space curve singularity is locally complete intersection, one obtains a relation between the latter and Euler numbers of Hilbert schemes of the space curve singularity. It is also shown that this relation yields explicit results for a class of torus-invariant locally complete intersection singularities.
Paper Structure (50 sections, 120 theorems, 491 equations)

This paper contains 50 sections, 120 theorems, 491 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Strategy of the proof
  4. Explicit examples
  5. Possible generalizations
  6. Open questions in low-dimensional topology
  7. Open questions in combinatorics and representation theory
  8. Outline
  9. Geometric setup
  10. $(0,-2)$ curves on threefolds
  11. Numerical classes
  12. Serre duality
  13. Rigidity and filtrations
  14. Semistable sheaves on the exceptional locus
  15. Semistable sheaves and filtrations
  16. ...and 35 more sections

Key Result

Theorem A

Let us consider a flop transition of the form eq:flop-diagram-A satisfying Assumptions assumption:f, assumption:divisor, and assumption:surface. Let $C\subset W$ be a reduced irreducible plane curve so that $\nu \in C^{\mathsf{sing}}$ and let $C^\pm \subset S^\pm$ be its strict transforms. Then, the

Theorems & Definitions (228)

  • Theorem A
  • Remark
  • Theorem B
  • Remark
  • Theorem C: Theorem \ref{['thm:C-framed-identity']}
  • Remark
  • Theorem D
  • Theorem E: Theorem \ref{['thm:lci-curves']}
  • Remark
  • Definition 2.1
  • ...and 218 more