ACM vector bundles on prime Fano threefolds and complete intersection Calabi Yau threefolds

TL;DR

This paper classifies all indecomposable rank-two vector bundles without intermediate cohomology on prime Fano and Calabi-Yau threefolds, linking their existence to specific curves with certain invariants.

Contribution

It provides a complete list of such vector bundles and relates their existence to the presence of curves with prescribed properties on these threefolds.

Findings

List of all possible indecomposable normalized rank-two vector bundles without intermediate cohomology.

Reduction of vector bundle existence problem to the existence of certain curves.

Known cases where such curves exist on general threefolds.

Abstract

In this paper we derive a list of all the possible indecomposable normalized rank--two vector bundles without intermediate cohomology on the prime Fano threefolds and on the complete intersection Calabi Yau threefolds, say $V$, of Picard number $\rho=1$. For any such bundle $\E$, if it exists, we find the projective invariants of the curves $C \subset V$ which are the zero-locus of general global sections of $\E$. In turn, a curve $C \subset V$ with such invariants is a section of a bundle $\E$ from our lists. This way we reduce the problem for existence of such bundles on $V$ to the problem for existence of curves with prescribed properties contained in $V$. In part of the cases in our lists the existence of such curves on the general $V$ is known, and we state the question about the existence on the general $V$ of any type of curves from the lists.