Quintic del Pezzo threefolds in positive and mixed characteristic
Tetsushi Ito, Akihiro Kanemitsu, Teppei Takamatsu, Yuuji Tanaka
TL;DR
This work extends the classical theory of quintic del Pezzo threefolds ($V_5$-varieties) from characteristic-zero, algebraically closed fields to arbitrary base schemes, establishing a classification by $B$-similarity classes of non-degenerate ternary symmetric bilinear forms, via an equivalence with nets of quinary alternating forms. It provides explicit geometric descriptions of automorphism group schemes and the Hilbert scheme of lines, and reveals new phenomena in characteristic two, including non-reduced automorphism groups and richer line-orbit structures. The paper then leverages this classification to obtain arithmetic finiteness results: good reduction criteria, explicit Shafarevich-type counts, and finiteness of integral models, including the striking fact that there are exactly two isomorphism classes of $V_5$-schemes over ${f Z}$. A key geometric insight is that $V_5$-schemes arise as the variety of trisecant lines to projected Veronese surfaces, a viewpoint that unifies the moduli and arithmetic perspectives. The results bridge moduli theory, arithmetic geometry, and the geometry of prime Fano threefolds, with additional applications to genus-$12$ Fano varieties and a pathway to a broader two-class classification over rings like ${f Z}$.
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.