On the existence of certain weak Fano threefolds of Picard number two

TL;DR

This paper investigates the existence of specific smooth weak Fano threefolds with Picard number two, focusing on those with small anti-canonical maps and previously classified invariants, excluding 12 cases.

Contribution

It establishes the existence or non-existence of certain weak Fano threefolds with Picard number two, completing the classification for most cases.

Findings

Confirmed existence for most numerical cases

Identified 12 cases with unresolved existence questions

Advances classification of weak Fano threefolds

Abstract

This article settles the question of existence of smooth weak Fano threefolds of Picard number two with small anti-canonical map and previously classified numerical invariants obtained by blowing up certain curves on smooth Fano threefolds of Picard number 1 with the exception of 12 numerical cases.