Compactifications of affine homology $3$-cells into blow-ups of the projective $3$-space with trivial log canonical divisors

TL;DR

This paper classifies all ways to compactify affine homology 3-cells into certain blow-ups of projective 3-space with trivial log canonical divisors, showing most are isomorphic to affine 3-space.

Contribution

It provides a complete classification of such compactifications, revealing that almost all are isomorphic to affine 3-space with one exception.

Findings

All compactifications are classified explicitly.

Most compactifications are isomorphic to affine 3-space.

A unique non-isomorphic example is identified.

Abstract

In this paper we classify all the compactifications of affine homology $3$-cells into the blow-ups of the projective $3$-space along smooth curves such that the log canonical divisors are linearly trivial. As a result, we prove that each embedded affine $3$-fold is isomorphic to the affine $3$-space except one example.