Fano threefolds in positive characteristic I

TL;DR

This paper classifies certain smooth Fano threefolds over algebraically closed fields of positive characteristic and proves properties about their embeddings, focusing on anti-canonical systems and intersections of quadrics.

Contribution

It provides a classification of smooth Fano threefolds with specific anti-canonical properties in positive characteristic and establishes new results on their embeddings.

Findings

Classification of smooth Fano threefolds with non-very ample anti-canonical systems

Proof that high-genus anti-canonically embedded Fano threefolds are intersections of quadrics

Insights into the geometry of Fano threefolds in positive characteristic

Abstract

Over an algebraically closed field of positive characteristic, we classify smooth Fano threefolds of Picard number one whose anti-canonical linear systems are not very ample. Furthermore, we also prove that an anti-canonically embedded Fano threefold of genus at least five is an intersection of quadrics.