On projective threefolds with two-dimensional space of vanishing cycles

TL;DR

This paper classifies certain smooth projective threefolds with a two-dimensional space of vanishing cycles and also characterizes rank 2 very ample vector bundles with specific Chern class on surfaces.

Contribution

It provides a complete classification of threefolds with two-dimensional vanishing cycles and rank 2 vector bundles with $c_2=3$ on surfaces.

Findings

List of all such threefolds identified

Complete classification of rank 2 very ample vector bundles with $c_2=3$

Insights into the geometry of vanishing cycles and vector bundles

Abstract

We obtain a complete list of smooth projective threefolds over $\mathbb C$ for which the dimension of the space of vanishing cycles (in $H^2$ of the smooth hyperplane section) equals $2$. We also obtain a complete list of rank 2 very ample vector bundles $E$ on smooth projective surfaces with $c_2(E) = 3$.