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Rank Two Sheaves With Low Discriminant on the Fano Threefold of Index 2 and Degree 5

Danil A. Vassiliev

TL;DR

This work determines sharp bounds for the third Chern character of rank-$2$ Gieseker semistable sheaves on the Fano threefold $X_5$ with low discriminant $\overline{\Delta}_H(E)\le 40$, using tilt-stability and Bridgeland stability in the derived category. It provides a concrete description of all such sheaves with maximal $c_3$, distinguishing cases by $c_1(E)=-1$ and $c_1(E)=0$, and identifies equality scenarios realized by standard bundles ($\mathcal{U}$, $\mathcal{O}_X(-1)$, $\mathcal{Q}(-1)$) and explicit extension constructions. The main results are established through a detailed analysis of tilt-stability, walls, and exceptional collections, relating rank-$2$ objects to four-term complexes in a Bridgeland heart $\mathfrak{C}$. The paper also conjectures a broader description for large discriminant, proposing that maximal $c_3$-type sheaves arise as extensions $0\to F\to E\to G\to 0$ with $F\in\{\mathcal{O}_X(-1)^{\oplus 2},\mathcal{U}\}$ and $G$ supported on a smooth hyperplane section, with supporting evidence built from Grothendieck–Riemann–Roch calculations on pushforwards of hyperplane sections and known moduli components. Overall, the results illustrate how Bridgeland stability and derived-category techniques yield concrete, extension-based descriptions of extremal rank-$2$ sheaves on a Fano threefold.

Abstract

We describe rank 2 Gieseker semistable sheaves $E$ on the Fano threefold $X_5$ of index 2 and degree 5 with maximal third Chern class $c_3(E)$ for all possible low values of discriminant $\overlineΔ_H(E)\le 40$. The work uses the theory of tilt-stability and Bridgeland stability conditions on smooth projective threefolds. We also make a conjecture about the rank 2 sheaves on $X_5$ with maximal $c_3$ and discriminant big enough.

Rank Two Sheaves With Low Discriminant on the Fano Threefold of Index 2 and Degree 5

TL;DR

This work determines sharp bounds for the third Chern character of rank- Gieseker semistable sheaves on the Fano threefold with low discriminant , using tilt-stability and Bridgeland stability in the derived category. It provides a concrete description of all such sheaves with maximal , distinguishing cases by and , and identifies equality scenarios realized by standard bundles (, , ) and explicit extension constructions. The main results are established through a detailed analysis of tilt-stability, walls, and exceptional collections, relating rank- objects to four-term complexes in a Bridgeland heart . The paper also conjectures a broader description for large discriminant, proposing that maximal -type sheaves arise as extensions with and supported on a smooth hyperplane section, with supporting evidence built from Grothendieck–Riemann–Roch calculations on pushforwards of hyperplane sections and known moduli components. Overall, the results illustrate how Bridgeland stability and derived-category techniques yield concrete, extension-based descriptions of extremal rank- sheaves on a Fano threefold.

Abstract

We describe rank 2 Gieseker semistable sheaves on the Fano threefold of index 2 and degree 5 with maximal third Chern class for all possible low values of discriminant . The work uses the theory of tilt-stability and Bridgeland stability conditions on smooth projective threefolds. We also make a conjecture about the rank 2 sheaves on with maximal and discriminant big enough.

Paper Structure

This paper contains 4 sections, 34 theorems, 65 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Derived Category $\mathrm D^b(X)$ and Stability
  3. Objects with Maximal Third Chern Character
  4. Partial Results and a Conjecture

Key Result

Theorem 1.1

Let $E$ be a Gieseker semistable sheaf of rank 2 on $X$ with Chern classes $c_1,c_2,c_3$. (1) If $c_1=-1$, then $c_2\ge 2$. (1.1) If $c_2=2$, then $c_3\le 0$. In case of equality we have $E\cong\mathcal{U}$. (1.2) If $c_2=3$, then $c_3\le 1$. In case of equality $E$ is included into an exact triple (2) If $c_1=0$, then $c_2\ge 0$. (2.1) If $c_2=0$, then $c_3\le 0$. In case of equality $E\cong\mat

Figures (1)

  • Figure 1: Curves in which some of $Z_{\alpha,\beta,\frac{1}{6}}(\mathcal{O}_{X}(-1)),Z_{\alpha,\beta,\frac{1}{6}}(\mathcal{Q}(-1)),Z_{\alpha,\beta,\frac{1}{6}}(\mathcal{U}),Z_{\alpha,\beta,\frac{1}{6}}(\mathcal{O}_{X})$ are $\mathbb R$-linearly dependent. The horizontal line is the $\beta$-axis, the vertical one is the $\alpha$-axis.

Theorems & Definitions (64)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3: Numerical properties of walls
  • proof
  • Proposition 2.4: Properties of actual walls
  • proof
  • Lemma 2.5
  • proof
  • Remark 2.6
  • ...and 54 more