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Moduli of rank two semistable sheaves on rational Fano threefolds of the main series

Alexander S. Tikhomirov, Danil A. Vassiliev

TL;DR

The article advances the study of moduli of semistable rank two sheaves on the main-series rational Fano threefolds by combining tilt/Bridgeland stability in the derived category with geometric constructions. It proves sharp $c_3$ bounds for semistable objects on $X_2$, and provides a complete description of moduli spaces with maximal $c_3$, including explicit extensions and reflexive cases; it also constructs several infinite series of moduli components on $X_1,X_2,X_4,X_5$, yielding numerous rational components and, in one case, an irrational family. Furthermore, it proves boundedness of $c_3$ for stable reflexive sheaves of general type with $c_1=0$ on $X_4$ and $X_5$, linking $c_2$-growth to quadratic bounds on $c_3$. Overall, the work substantially expands the landscape of rank two moduli on Fano threefolds, clarifying structure, rationality, and the role of reflexive general-type objects in the global geometry of these moduli spaces.

Abstract

In this paper we investigate the moduli spaces of semistable coherent sheaves of rank two on the projective space $\mathbb{P}^3$ and the following rational Fano manifolds of the main series - the three-dimensional quadric $X_2$, the intersection of two 4-dimensional quadrics $X_4$ and the Fano manifold $X_5$ of degree 5. For the quadric $X_2$, the boundedness of the third Chern class $c_3$ of rank two semistable objects in $\mathrm{D}^b(X_2)$, including sheaves, is proved. An explicit description is given of all the moduli spaces of semistable sheaves of rank two on $X_2$, including reflexive ones, with a maximal third class $c_3\ge0$. These spaces turn out to be irreducible smooth rational manifolds in all cases, except for the following two: $(c_1,c_2,c_3)=(0,2,2)$ or (0,4,8). Several new infinite series of rational components of the moduli spaces of semistable sheaves of rank two on $\mathbb{P}^3$, $X_2$, $X_4$ and $X_5$ are constructed, as well as a new infinite series of irrational components on $X_4$. The boundedness of the class $c_3$ is proved for $c_1=0$ and any $c_2>0$ for stable reflexive sheaves of general type on manifolds $X_4$ and $X_5$.

Moduli of rank two semistable sheaves on rational Fano threefolds of the main series

TL;DR

The article advances the study of moduli of semistable rank two sheaves on the main-series rational Fano threefolds by combining tilt/Bridgeland stability in the derived category with geometric constructions. It proves sharp bounds for semistable objects on , and provides a complete description of moduli spaces with maximal , including explicit extensions and reflexive cases; it also constructs several infinite series of moduli components on , yielding numerous rational components and, in one case, an irrational family. Furthermore, it proves boundedness of for stable reflexive sheaves of general type with on and , linking -growth to quadratic bounds on . Overall, the work substantially expands the landscape of rank two moduli on Fano threefolds, clarifying structure, rationality, and the role of reflexive general-type objects in the global geometry of these moduli spaces.

Abstract

In this paper we investigate the moduli spaces of semistable coherent sheaves of rank two on the projective space and the following rational Fano manifolds of the main series - the three-dimensional quadric , the intersection of two 4-dimensional quadrics and the Fano manifold of degree 5. For the quadric , the boundedness of the third Chern class of rank two semistable objects in , including sheaves, is proved. An explicit description is given of all the moduli spaces of semistable sheaves of rank two on , including reflexive ones, with a maximal third class . These spaces turn out to be irreducible smooth rational manifolds in all cases, except for the following two: or (0,4,8). Several new infinite series of rational components of the moduli spaces of semistable sheaves of rank two on , , and are constructed, as well as a new infinite series of irrational components on . The boundedness of the class is proved for and any for stable reflexive sheaves of general type on manifolds and .
Paper Structure (14 sections, 32 theorems, 199 equations)

This paper contains 14 sections, 32 theorems, 199 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Semistable objects of rank two in $D^b(X_2)$ with maximal third Chern class $c_3$
  4. Infinite series of components of moduli spaces of semistable sheaves of rank two on varieties $X_1$, $X_2$, $X_4$ and $X_5$
  5. First series.
  6. Second series.
  7. Third series.
  8. Reflexive stable sheaves of general type with $c_1=0$ on $X_4$ and $X_5$.
  9. Moduli spaces of rank two semistable sheaves with maximal third Chern class on the three-dimensional quadric $X_2$
  10. General case of rank two semistable sheaves with $-1\le c_1\le0$ and maximal class $c_3$ on the quadric $X_2$.
  11. Special cases of rank two semistable sheaves with small values of class $c_2$ and maximal class $c_3\ge0$ on $X_2$.
  12. Boundedness of the third Chern class of rank two stable reflexive sheaves of general type with $c_1=0$ on the varieties $X_4$ and $X_5$
  13. Rank two stable reflexive sheaves of general type with $c_1=0$ on the variety $X_5$
  14. Rank two stable reflexive sheaves of general type on $X_4$

Key Result

Proposition 2.1

(i) An object $E\in\operatorname{Coh}^{\beta}(X)$ is $\nu_{\alpha,\beta}$-(semi)stable for $\beta<\mu(E)$ and $\alpha\gg0$ iff $E$ is a 2-(semi)stable sheaf. (ii) Let $X=X_2$. For any $\nu_{\alpha,\beta}$-semistable object $E\in\operatorname{Coh}^\beta(X_2)$ with $\ch(E)=(r,cH,dH^2,e)$ we have the f

Theorems & Definitions (56)

  • Proposition 2.1
  • proof
  • Proposition 2.2: Sch18
  • Lemma 2.1
  • Lemma 2.2: BMT
  • Proposition 2.3: MS18
  • Proposition 2.4: Sch14,Sch19
  • Remark 2.1
  • Theorem 3.1
  • Remark 3.2
  • ...and 46 more