On the moduli of degree 4 Del Pezzo surfaces

On stable rationality of Fano threefolds and del Pezzo fibrations

“…Here S → P 1 is a rational ruled surface. The conic bundle structure over S yields a discriminant curve D ⊂ S and an étale double cover D ′ → D. Note that D ′ → D coincides with the spectral data introduced in [HKT15a, § §2,8] and S is the natural ruled surface containing D described in [HKT15a,§10].…”

“…), an even integer (see [HKT15a,HT14] for more background). The principal results we require are [HKT15a, Th.…”

“…• when h(X ) = 8 or 10, the moduli space of these fibrations has two irreducible components; • when h(X ) ≥ 12 the moduli space is irreducible. When h(X ) = 8, 10, 12 the total space X is either rational or birational to a cubic threefold; see [HKT15a,§11] and [HT14, §8-10] for details. Thus we will focus on fibrations with heights at least fourteen.…”

“…10.1] implies that the D ⊂ S arising from del Pezzo fibrations are generic in the linear series L = |D|. The analysis of [HKT15a,§3] shows that the monodromy acts on H 1 (D, Z/2Z) via the full symplectic group, hence transitively on the non-trivial elements. To apply the main result of Section 3, it suffices to exhibit a distinguished point in |D|, i.e., a reducible curve D = D 1 ∪ D 2 with D 1 and D 2 smooth of positive genus, intersecting transversally.…”

“…Nevertheless, conic bundles are a useful tool for analyzing stable rationality of Fano threefolds. Second, in [HKT15a] we classified quartic del Pezzo surfaces with mild singular fibers and maximal monodromy; previously [HT14] we showed that a number of these arise as specializations of Fano threefolds of index one. Together, these facilitate a streamlined approach to most families of non-rational Fano threefolds.…”