Quintic del Pezzo threefolds in positive and mixed characteristic

TL;DR

This paper classifies smooth quintic del Pezzo threefolds over various bases using symmetric bilinear forms, explores their automorphisms and lines, and reveals new phenomena in characteristic two, with implications for arithmetic geometry.

Contribution

It provides a classification of quintic del Pezzo threefolds via bilinear forms and analyzes their automorphisms, lines, and orbit structures, including new phenomena in characteristic two.

Findings

Classification by non-degenerate ternary symmetric bilinear forms

Description of automorphism group schemes and Hilbert schemes of lines

Identification of new phenomena in characteristic two

Abstract

We show that smooth quintic del Pezzo threefolds over arbitrary base schemes are classified by non-degenerate ternary symmetric bilinear forms. Then we describe the automorphism group schemes, the Hilbert schemes of lines and the orbit structures of quintic del Pezzo threefolds, and we find several new phenomena in characteristic two. As arithmetic applications, we prove a refinement of the Shafarevich conjecture, and prove that there are exactly two isomorphism classes of quintic del Pezzo threefolds over the ring of rational integers.