Involution surface bundles over surfaces

Abstract

We construct models of involution surface bundles over algebraic surfaces, degenerating over normal crossing divisors, and with controlled singularities of the total space.

Models of quadric surface bundles

“…Here we show that the third step succeeds, in the sense that the direct image of the locally free coherent sheaf remains locally free; it is enough to check this étale locally and therefore it suffices to work with étale local models as in [19,Section 3]. But we will see that the fourth step leads, generally, to a scheme that is not flat over the base.…”

“…Let k be a perfect field of characteristic different from 2. Having (see reduction steps) achieved a finite degree 2 covering of smooth varieties with associated finite étale covering of root stacks and smooth P 1 -fibration over the source root stack determining a smooth P 1 × P 1 -fibration over the target root stack, one might ask what kind of fibration arises by applying the construction from [19], that is:…”

“…Another point of view, taken in [4], takes as its starting point the embedding of a quadric surface by its anticanonical linear system as a degree 8 del Pezzo surface in P 8 , or involution surface; an involution surface is isomorphic to a quadric surface if and only if it admits a line bundle whose class is half the anticanonical class. Over a surface base, involution surface bundles are described in [19].…”

“…5 the étale local form is justified, completing the proof of Theorem 1.1. Appendix A shows that the construction of involution surface bundles in [19] may lead to a non-flat model when attempted over a base of dimension > 2.…”

Models of quadric surface bundles

“…In Section 5 the étale local form is justified, completing the proof of Theorem 1. Appendix A shows that the construction of involution surface bundles in [19] may lead to a non-flat model when attempted over a base of dimension > 2.…”