Vector bundles on bielliptic surfaces: Ulrich bundles and degree of irrationality

Technical Summary: Vector Bundles on Bielliptic Surfaces: Ulrich Bundles and Degree of Irrationality

Problem Statement
This paper addresses two distinct but interconnected problems regarding the geometry of bielliptic surfaces $S$:

1. Classification of Ulrich Bundles: Determining the Chern characters (specifically, the Mukai vectors) of Ulrich bundles on bielliptic surfaces. While the existence of rank 2 Ulrich bundles on surfaces of Kodaira dimension zero is established, a complete classification of their Chern characters depending on the topological type of the surface remains an open problem.
2. Degree of Irrationality: Computing the degree of irrationality, $\text{irr}(S)$, defined as the minimal degree of a generically finite rational map from $S$ to $\mathbb{P}^2$. Previous work by Yoshihara [31] computed this invariant for all bielliptic surface types except Type 6.

Methodology
The author employs a combination of moduli space theory, derived category techniques, and classical algebraic geometry.

• Weak Brill-Noether Property: A central tool is the investigation of the weak Brill-Noether property for moduli spaces of sheaves $M_H(v)$. A moduli space satisfies this property if a general sheaf in it has at most one non-zero cohomology group. The paper establishes this property for line bundles on bielliptic surfaces, distinguishing between cases where the torsion subgroup $H^2(S, \mathbb{Z})_{\text{tor}}$ is non-trivial (Types 1, 2, 3, 5) and where it is trivial (Types 4, 6, 7).
• Moduli Spaces and Isotropic Vectors: The study of Ulrich bundles is reduced to analyzing moduli spaces of stable sheaves with isotropic Mukai vectors. The author leverages recent existence results by Nuer [25] regarding the non-emptiness of these moduli spaces for generic polarizations.
• Canonical Covers and Étale Covers: The analysis frequently utilizes the canonical cover $\tilde{X}$ (an abelian surface) and intermediate étale covers to lift problems from $S$ to a setting where the geometry is better understood (e.g., using properties of semi-homogeneous sheaves on abelian varieties).
• Degree of Irrationality via Stable Bundles: To compute $\text{irr}(S)$, the paper adapts a method developed by Moretti [21] for K3 surfaces. This involves constructing a "good pair" $(E, V)$, where $E$ is a stable rank 2 vector bundle and $V \subset H^0(S, E)$ generates $E$ in codimension 2. The degree of the induced rational map $\phi_V: S \dashrightarrow \mathbb{P}^2$ is then calculated using the formula $\deg(\phi_V) = c_2(E) - \deg(\bigcap Z(s))$.

Key Contributions and Results

1. Classification of Ulrich Bundles (Theorem 1.3)
The paper provides a complete classification of the Mukai vectors of Ulrich bundles on bielliptic surfaces polarized by a generic ample divisor $H$ of numerical class $aA_0 + bB_0$ (with $b \ge \lambda_S$). The classification depends strictly on the topological type of $S$:

• Types 1, 2, 3, 5: These surfaces carry Ulrich bundles for all ranks $r \ge 1$ with Mukai vectors $v_{\text{Ulrich}}(r, k) = (r, ka_0 + (3r-k)bB_0, 2rab)$ for all integers $r \le k \le 2r$. This includes the existence of Ulrich line bundles.
• Types 4, 6, 7: These surfaces carry Ulrich bundles of every rank $r \ge 2$, but they do not carry Ulrich line bundles. Furthermore, not all theoretical Mukai vectors $v_{\text{Ulrich}}(r, k)$ correspond to actual Ulrich bundles; specific obstructions arise due to the vanishing of the torsion subgroup $H^2(S, \mathbb{Z})_{\text{tor}}$. For instance, Type 4 surfaces lack Ulrich bundles of Mukai vector $v_{\text{Ulrich}}(3, 3)$, and Type 6 surfaces lack those of $v_{\text{Ulrich}}(2, 2)$.

2. Degree of Irrationality (Theorem 1.5)
The paper proves that for all bielliptic surfaces $S$, the degree of irrationality satisfies $\text{irr}(S) \le 3$.

• This result completes the work of Yoshihara [31], specifically resolving the previously open case of Type 6 surfaces.
• The proof constructs a specific stable rank 2 vector bundle $E$ with Mukai vector $(2, D, 3)$ where $D$ has numerical class $3A_0 + 2B_0$. By demonstrating that $E$ is globally generated in codimension 2, the author establishes a generically finite rational map of degree at most 3.
• The strategy confirms that the bound $\text{irr}(S) \le 3$ holds uniformly across all seven families of bielliptic surfaces.

Significance and Claims
The paper claims to provide the first complete classification of Ulrich bundles on bielliptic surfaces, highlighting how the topological invariants (specifically the torsion in the second cohomology group) dictate the existence of such bundles. It demonstrates that the weak Brill-Noether property for line bundles is a critical determinant in the existence of Ulrich line bundles.

Regarding the degree of irrationality, the paper claims to settle the question for bielliptic surfaces by providing a uniform treatment that adapts Moretti's method from K3 surfaces. The result $\text{irr}(S) \le 3$ is presented as a definitive upper bound for this class of surfaces, filling the gap left by previous literature. The work relies on the interplay between the geometry of the canonical cover and the stability conditions of sheaves on the base surface, offering a concrete application of moduli space theory to birational invariants.