Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Modular Serre Correspondence via stable pairs

Marcos Jardim, Dapeng Mu

Abstract

A stable pair on a projective variety consists of a sheaf and a global section subject to stability conditions parameterized by rational polynomials. We will show that for a smooth projective threefold and a class of a rank 2 sheaf, there are two stability chambers (in the space of rational polynomials under the lexicographic order) for which the moduli spaces of semistable pairs admit morphisms to a Gieseker moduli space of rank 2 semistable sheaves and a Hilbert scheme, respectively. In the latter moduli space, every semistable pair corresponds to a closed sub-scheme of codimension 2 with an extension class, providing a generalization of the Serre correspondence. These two moduli spaces are related by finitely many wall-crossings. We provide explicit descriptions of those wall-crossings for certain fixed numerical classes. In particular, these wall-crossings preserve the connectedness of the moduli space of semistable pairs.

Modular Serre Correspondence via stable pairs

Abstract

A stable pair on a projective variety consists of a sheaf and a global section subject to stability conditions parameterized by rational polynomials. We will show that for a smooth projective threefold and a class of a rank 2 sheaf, there are two stability chambers (in the space of rational polynomials under the lexicographic order) for which the moduli spaces of semistable pairs admit morphisms to a Gieseker moduli space of rank 2 semistable sheaves and a Hilbert scheme, respectively. In the latter moduli space, every semistable pair corresponds to a closed sub-scheme of codimension 2 with an extension class, providing a generalization of the Serre correspondence. These two moduli spaces are related by finitely many wall-crossings. We provide explicit descriptions of those wall-crossings for certain fixed numerical classes. In particular, these wall-crossings preserve the connectedness of the moduli space of semistable pairs.
Paper Structure (43 sections, 41 theorems, 210 equations, 15 figures)

This paper contains 43 sections, 41 theorems, 210 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Coherent pairs
  3. Basic facts about coherent pairs
  4. Stability of pairs vs. stability of sheaves
  5. Critical values (walls) for $\delta$
  6. Special walls for a rank 2 class
  7. The wall for saturated pairs
  8. The first wall from the left
  9. Moduli spaces of coherent pairs
  10. Moduli space of stable pairs
  11. Deformation theory
  12. Wall-crossings for a rank 2 class
  13. General wall-crossings at critical values
  14. Wall-crossing phenomena
  15. Description of walls for rank 2 pairs
  16. ...and 28 more sections

Key Result

Lemma 2.2

If $\Lambda=(E, s)$ and $\Lambda'=(E', s')$ are two coherent pairs on $X$, then there is a long exact sequence as follows Here, $\mathbb{C}\cdot(s)$ denotes the $\mathbb{C}$-vector space over by the section $s$.

Figures (15)

  • Figure 1.1: Wall-crossings for a rank 2 class.
  • Figure 1.2: Modular Serre correspondence, where ${\mathcal{P}}$ is an irreducible component of ${\mathcal{S}}^{\delta}_{X,L}(v)$ whose general pair is both saturated and very stable. Here, $\Gamma(E,s)=(L/\mathop{\mathrm{coker}}\nolimits(s))\otimes L^\vee$, while $\Psi(E,s)=E$.
  • Figure 1.3: 0-dimensional walls
  • Figure 3.1: Gieseker-Hilbert
  • Figure 3.2: Modular Serre correspondence
  • ...and 10 more figures

Theorems & Definitions (99)

  • Remark 2.1
  • Lemma 2.2: he1996espaces Corollary 1.6
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • ...and 89 more