Quartic del Pezzo surfaces over function fields of curves

On the moduli of degree 4 Del Pezzo surfaces

“…In Section 10 we state and prove our main theorems, describing and enumerating the components of general families of degree 4 Del Pezzo surfaces over P 1 . In an Appendix, we show that the discrete invariant of families introduced here agrees with the height defined in [18].…”

“…The morphism (6.3) restricts to (6.4) M → [U/PGL 2 ], also a neutral étale gerbe. However, even though [U/PGL 2 ] is separated (and, in fact, is proper), the stack M is nonseparated, since as remarked in the Introduction (see also [18,Rem. 2]), the order of the geometric stabilizer group in a family may decrease under specialization.…”

“…By Proposition 13, the degrees in (6.8) must be related by a constant proportionality. We deduce their equality from any of the worked out examples, e.g., Case 1 on page 11 of [18] with…”

“…Height 8: We recall the construction from [18,Remark 15]. Let X ⊂ P 1 × P 5 be a complete intersection of a form of bidegree (1, 1) and two forms of bidegree (0, 2).…”

“…If we assume that family is generically smooth with square-free discriminant, then X is a smooth projective threefold. In this case in [18] the height is defined as a triple intersection number on X . Here we show that these two definitions agree.…”