Diophantus Revisited: On rational surfaces and K3 surfaces in the Arithmetica

TL;DR

This paper explores how Diophantus' problems relate to modern algebraic surfaces, showing that some lead to rational surfaces and others to K3 surfaces, with Diophantus' solutions providing modern parametrizations.

Contribution

It reveals the connection between ancient Diophantine problems and complex algebraic surfaces, offering new insights into their geometric structures and parametrizations.

Findings

Certain Diophantus problems correspond to rational surfaces.

Other problems lead to K3 surfaces.

Diophantus' solutions yield modern parametrizations.

Abstract

This article wants to show two things: first, that certain problems in Diophantus' Arithmetica lead to equations defining del Pezzo surfaces or other rational surfaces, while certain others lead to K3 surfaces; second, that Diophantus' own solutions to these problems, when viewed through a modern lens, lead to parametrizations of these surfaces, or of parametrizations of rational curves lying on them.