Stable rank 3 vector bundles on $\mathbb{P}^3$ with $c_1 = 0$, $c_2 = 3$

TL;DR

This paper investigates the restriction properties of stable rank 3 vector bundles with zero first Chern class and second Chern class three on projective 3-space, revealing more exceptions than previously conjectured and analyzing their moduli spaces.

Contribution

It clarifies the previously undecided case for c2=3, identifying new exceptions to stable restriction and describing the properties of their moduli spaces.

Findings

More exceptions to stable restriction than previously conjectured.

Identification of a Schwarzenberger bundle with c3=6 as an exception.

Moduli spaces are nonsingular, connected, and 28-dimensional.

Abstract

We clarify the undecided case $c_2 = 3$ of a theorem of Ein, Hartshorne and Vogelaar [Math. Ann. 259 (1982), 541--569] about the restriction of a stable rank 3 vector bundle with $c_1 = 0$ on the projective 3-space to a general plane. It turns out that there are more exceptions to the stable restriction property than those conjectured by the three authors. One of them is a Schwarzenberger bundle (twisted by $-1$); it has $c_3 = 6$. There are also some exceptions with $c_3 = 2$ (plus, of course, their duals). We also prove, for completeness, the basic properties of the corresponding moduli spaces; they are all nonsingular and connected, of dimension 28.