A note on the moduli of stable sheaves on elliptic ruled surfaces

1. Problem Statement

The paper addresses the geometry of moduli spaces of $\mu$-semistable coherent sheaves on elliptic ruled surfaces (also known as $\mathbb{P}^1$-bundles over a smooth projective curve $C$).

• Context: Let $\pi: X \to C$ be a $\mathbb{P}^1$-bundle over a curve of genus $g$. Let $C_0$ be a minimal section with self-intersection $(C_0^2) = -e$, and $f$ be a fiber. The canonical divisor is $K_X \equiv -2C_0 - (2-2g+e)f$.
• Previous Work: Walter [7] established the irreducibility of these moduli spaces under the strict condition $((K_X + f) \cdot H) < 0$. This condition implies that sheaves are "prioritary," allowing the use of specific deformation theories.
• The Gap: The author aims to generalize these results to the broader condition $(K_X \cdot H) < 0$.
  • This condition holds for any ample divisor $H$ if $g=0$ or $g=1$.
  • Specifically, the author focuses on the cases where $g=1$ and $e=0$ or $e=-1$ (where $-K_X$ is nef but not necessarily ample in the sense of Walter's original condition).
• Core Questions:
  1. Are the moduli spaces $\mathcal{M}_H(e)^{\mu ss}$ irreducible under the weaker condition $(K_X \cdot H) < 0$?
  2. What are the necessary and sufficient conditions for the existence of such sheaves (non-emptiness)?
  3. What are the properties of the general members of these moduli spaces (e.g., local freeness, stability on fibers)?

2. Methodology

Yoshio employs a combination of stack theory, deformation theory, and wall-crossing analysis within the ample cone.

• Moduli Stacks: Instead of using equivariant cohomology of Quot-schemes (as in previous works), the author utilizes moduli stacks of coherent sheaves ($M(e)$, $\mathcal{M}_H(e)^{\mu ss}$). This allows for a more direct handling of families and stability conditions.
• Filtration Stacks: The author introduces the stack of filtrations $FH(e_1, \dots, e_s)$ to analyze the Harder-Narasimhan and Jordan-Hölder filtrations. This is crucial for dimension counting and comparing components of moduli spaces across different polarizations.
• Wall and Chamber Analysis:
  • The ample cone $\text{Amp}(X)$ is divided into chambers separated by walls defined by $W_F = \{H \mid ((\mu(E) - \mu(F)) \cdot H) = 0\}$.
  • The author analyzes how the moduli space changes as the polarization $H$ moves across these walls.
• Key Technical Tool: Vanishing of $Ext^2$:
  • A central lemma (Lemma 1.4) proves that if $(K_X \cdot H) < 0$, then for any two $\mu$-semistable sheaves $E, F$ with the same slope, $\text{Ext}^2(E, F) = 0$.
  • This vanishing implies that the moduli stack is smooth and allows the application of deformation theory to prove irreducibility.
• Blow-up Technique: To prove irreducibility of the stable locus, the author uses a result by Nakashima relating moduli spaces on $X$ to those on a blow-up $\tilde{X}$, applying Beauville's and Markman's theorems on surfaces with $(K_X \cdot H) < 0$.

3. Key Contributions and Results

A. Irreducibility (Theorem 0.1 & Proposition 2.7)

The paper proves that under the condition $(K_X \cdot H) < 0$, the moduli stack $\mathcal{M}_H(e)^{\mu ss}$ is irreducible for any Chern character $e = (r, \xi, \chi)$ satisfying $0 \le (\xi \cdot f) < r$.

• This generalizes Walter's result by removing the requirement that $((K_X + f) \cdot H) < 0$.
• The proof relies on showing that the locus of strictly semistable sheaves has strictly lower dimension than the stable locus (Lemma 2.5), and that the stable locally free locus is open and dense.

B. Existence Conditions (Theorem 0.1 & Proposition 2.9)

The author provides a precise criterion for the non-emptiness of the moduli space:
$$\mathcal{M}_H(e)^{\mu ss} \neq \emptyset \iff r^2 \Delta(e) \ge r_1(r - r_1)(x - e/2)$$
where:

• $H(x) = C_0 + xf$ is the ample divisor.
• $r_1 = (\xi \cdot f)$.
• $\Delta(e)$ is the discriminant defined in (1.2).
• For the specific case of genus $g=1$ and $e=0, -1$, the condition simplifies to $x \le \frac{e}{2} + \frac{r_2}{r_1 r_2} \Delta(e)$.

C. Properties of General Members (Theorem 0.1 (3))

For $r > 1$, a general member $E$ of the moduli space is:

1. A $\mu$-stable locally free sheaf.
2. Rigid on fibers: The restriction $E|_{\pi^{-1}(t)}$ is a rigid vector bundle for a general point $t \in C$.
3. This rigidity implies $\text{Ext}^1(E|_{\pi^{-1}(t)}, E|_{\pi^{-1}(t)}) = 0$.

D. Corollary for Nef Anticanonical Divisors (Corollary 0.2)

In the specific case where $-K_X$ is nef ($e=0, -1$), the moduli space is non-empty if and only if:

• $\Delta(e) > 0$, OR
• $r \mid (\xi \cdot f)$ and $\Delta(e) = 0$.

4. Significance

• Generalization of Irreducibility: The paper successfully extends the known irreducibility results for moduli spaces of sheaves on ruled surfaces from a restrictive condition involving the fiber $f$ to the more natural condition involving the canonical divisor $K_X$.
• Handling Genus 1: It specifically addresses the case of elliptic ruled surfaces ($g=1$) where the geometry is more subtle (e.g., when $e=0$ or $-1$), providing existence criteria that were previously unclear.
• Structural Insight: By utilizing the stack of filtrations and wall-crossing arguments, the paper clarifies the relationship between the stability of sheaves on the surface and the stability of their restrictions to fibers.
• Foundation for Further Study: The results on the non-emptiness and irreducibility of these moduli spaces provide a solid foundation for studying the geometry of these surfaces, potentially impacting the study of Donaldson invariants or the classification of vector bundles on elliptic surfaces.

In summary, Yoshio's note refines the understanding of moduli spaces on elliptic ruled surfaces by proving their irreducibility under a broader set of geometric conditions and establishing precise bounds for their existence, utilizing a robust framework of stack theory and deformation analysis.