A Finiteness Theorem for Del Pezzo Surfaces over Algebraic Number Fields

Néron models and the arithmetic Shafarevich conjecture for certain canonically polarized surfaces

“…In [1], André proved the arithmetic Shafarevich conjecture for polarized K3 surfaces and very polarized hyper-Kähler varieties (under some additional hypotheses). Moreover, Scholl proved the analogous finiteness statement for anti-canonically polarized surfaces (that is, del Pezzo surfaces) and Brauer-Severi varieties in [31]. It seems reasonable to suspect that finiteness results similar to those of André, Faltings and Scholl should be rife in arithmetic geometry.…”

Effectively Computing Integral Points on the Moduli of Smooth Quartic Curves

“…A smooth del Pezzo surface over a field k is split, or standard, if all lines are defined over k (see [37]).…”

“…In [37] Scholl proved the finiteness of all smooth del Pezzo surfaces over a number field K with good reduction outside a fixed set of finite places S of K. In particular, the set of O K,S -isomorphism classes of smooth del Pezzo surfaces over O K,S of degree at most four is finite. To prove Theorem 1.1, we establish an effective version of the latter finiteness statement.…”

Good reduction of algebraic groups and flag varieties

“…Theorem 1.4 generalises some already known results. Namely, the analogue of Theorem 1.4 was already known for Brauer-Severi varieties [17,Thm. 5.3] and quadric hypersurfaces in projective space [10].…”

“…The case of Fano varieties of dimension 1 (i.e. conics) is classical and the case of dimension 2 is dealt with in [17]. The case of flag varieties, handled in Theorem 1.4, is in some respects the next easiest case in this programme.…”