Ulrich wildness of some decomposable threefold scrolls over $\mathbb F_a$

Technical Summary: Ulrich Wildness of Some Decomposable Threefold Scrolls over $\mathbb{F}_a$

Problem Statement
The paper investigates the existence and structural properties of Ulrich bundles on threefold scrolls $X$ over Hirzebruch surfaces $\mathbb{F}_a$ ($a \geq 0$). While Ulrich bundles are well-characterized for curves and certain surfaces, their existence and rank classification remain largely open for varieties of dimension $n \geq 3$. Specifically, the authors address the "Ulrich wildness" of $X$, a property indicating that the variety supports arbitrarily large families of pairwise non-isomorphic, indecomposable Ulrich bundles.

Previous work [9] established that for $0 \leq a \leq b \leq 1$, the moduli spaces of Ulrich bundles on these scrolls contain generically smooth components arising from iterated extensions of specific line bundles. However, this earlier work was limited by modular obstructions in cases where $a$ and $b$ fall outside this range, particularly preventing the construction of smooth modular families for higher-rank bundles. This paper aims to resolve the remaining cases, distinguishing between scenarios where bundles are "inherited" from the base surface and those where "obstructions" prevent smooth moduli, yet wildness still holds.

Methodology
The authors analyze the threefold scroll $X = \mathbb{P}_{\mathbb{F}_a}(E^b_a)$, where $E^b_a = \mathcal{O}_{\mathbb{F}_a} \oplus \mathcal{O}_{\mathbb{F}_a}(0, -b)$, embedded via a very ample divisor $h = \mathcal{O}_a(1, 1, c)$ with $c \geq a + b + 1$. The methodology relies on constructing rank-$r$ Ulrich bundles via recursive extensions of specific line bundles and analyzing the dimension of the resulting extension spaces and moduli components.

The approach is divided into two primary regimes based on the parameters $a$ and $b$:

1. The Inherited Case ($b \geq 0 = a$):

   • The authors consider deformations of iterated extensions of the pair of line bundles $(L, L^U)$, which are twisted pullbacks from the base $\mathbb{F}_0$.
   • They construct rank-$r$ bundles $G_r$ via exact sequences $0 \to G_{r-1} \to G_r \to L_{\epsilon_r} \to 0$, alternating between $L$ and $L^U$.
   • To ensure stability and simplicity for $r \geq 3$, the authors replace the strictly semistable middle terms of the recursive construction with slope-stable bundles $U_{r-1}$ from the previously constructed modular component.
   • They compute Chern classes recursively and determine the dimension of the moduli space using the Euler characteristic of the endomorphism bundle.
   • Unirationality of the moduli components is established by relating the bundles on $X$ to Ulrich bundles on the base $\mathbb{F}_0$ via a morphism from the moduli space of bundles on $\mathbb{F}_0$, which is known to be unirational.

2. The Obstructed Case ($b \geq a \geq 0$, excluding specific small cases):

   • This regime involves the pair $(N, N^U)$, where cohomological obstructions (specifically non-vanishing $H^2$ groups) prevent the existence of generically smooth modular components for the iterated extensions.
   • Despite these obstructions, the authors prove Ulrich wildness by demonstrating the existence of families of indecomposable bundles.
   • They utilize an alternating extension strategy similar to the inherited case but work with bundles that remain strictly semistable.
   • Key to the proof is showing that for general extension classes, the resulting middle-term bundle is indecomposable. This is achieved by analyzing the action of the automorphism group $\text{Aut}(G_{r-1}) \times \text{Aut}(N_{\epsilon_r})$ on the extension space $\text{Ext}^1(G_{r-1}, N_{\epsilon_r})$.
   • By inductively computing dimensions of $\text{Hom}$ and $\text{End}$ groups, they show that the dimension of the extension space strictly exceeds the dimension of the orbit under the automorphism group action. This ensures that a general extension yields a non-isomorphic, indecomposable bundle.

Key Contributions and Results

• Main Theorem (Part 1): For the case $b \geq 0 = a$ and $c \geq b + 1$, the moduli space of rank-$r$ Ulrich bundles on $X$ contains a generically smooth, unirational component $M(r)$ for any $r \geq 2$.

  • The dimension of $M(r)$ is computed explicitly: $(r^2 - 1)(2c - b - 1)$ for odd $r$, and $r^2(2c - b - 1) + 1$ for even $r$.
  • The general point of $M(r)$ corresponds to a slope-stable Ulrich bundle.
  • Consequently, $X$ is shown to be geometrically $h$-Ulrich wild with no gaps in the ranks of slope-stable Ulrich bundles ($U_r(X) = \mathbb{N}^*$).

• Main Theorem (Part 2): For cases where $0 \leq a \leq 1, b \geq 2$ or $b \geq a \geq 1$ (excluding $(1,1)$), $X$ is geometrically $h$-Ulrich wild.

  • Although generically smooth modular components may not exist due to cohomological obstructions, the paper proves the existence of families of indecomposable Ulrich bundles of arbitrary rank $r \geq 2$.
  • This establishes that there are no gaps in the ranks of indecomposable Ulrich bundles ($U_r(X) = \mathbb{N}^*$), even in the presence of modular obstructions.

• Chern Class Formulas: The paper provides explicit recursive formulas for the Chern classes $c_1, c_2, c_3$ of the constructed rank-$r$ Ulrich bundles, distinguishing between odd and even ranks.

Significance
The paper extends the understanding of Ulrich bundles to higher-dimensional varieties where modular obstructions typically hinder classification. By distinguishing between "inherited" and "obstructed" cases, the authors demonstrate that the property of Ulrich wildness is robust; it persists even when the standard machinery for constructing smooth moduli spaces fails. The results confirm that decomposable threefold scrolls over Hirzebruch surfaces support rich internal geometries characterized by large families of indecomposable Ulrich bundles, filling a gap in the classification of Ulrich bundles for $n \geq 3$. The work relies on and extends the foundational results of [9], specifically addressing the cases where the parameters $a$ and $b$ exceed the previously studied range of $0 \leq a \leq b \leq 1$.