Arcs in finite projective spaces

TL;DR

This paper provides a comprehensive overview of large arcs in finite projective spaces, unifying and simplifying key theorems, including proofs of Segre's lemma and the MDS conjecture over prime fields.

Contribution

It offers a self-contained exposition with simplified proofs of important results, including the most general form of Segre's lemma of tangents and the MDS conjecture.

Findings

Unified proofs of classical and recent theorems on arcs

Proof of the most general form of Segre's lemma of tangents

Short proof of the MDS conjecture over prime fields

Abstract

This is an expository article detailing results concerning large arcs in finite projective spaces, which attempts to cover the most relevant results on arcs, simplifying and unifying proofs of known old and more recent theorems. The article is mostly self-contained and includes a proof of the most general form of Segre's lemma of tangents and a short proof of the MDS conjecture over prime fields based on this lemma.