Relative minimality for conic bundles

Question

Due to difficulties in finding references I am not sure of what relative minimality is about for conic bundles. Suppose we're working on con. bun. over surfaces.

The definition is ok: the fiber over any irreducible component of the discriminant curve should not be reducible. But how can I handle that concretely?

My impression is that there should always be somehow a way to reduce a non-relatively minimal conic bundle to a rel. minimal one (via birational maps, say blow-ups). Does this exist? Is there a canonical way? I would be glad if someone could make these guesses of mine more precise or give me some references.

Answer 1

You might consult

http://hal.archives-ouvertes.fr/docs/00/46/16/98/PDF/modelsgeomratcompositio.pdf

Your own definition seems a little confusing. A conic bundle is a map whose fibers are conics. Of course conics are embedded objects so this requires some kind of definition of a conic structure on the fibers. Plane conics come in three versions, irreducible and smooth, two distinct lines, and a double line. Usually a conic bundle has only the first two types and the locus in the target over which fibers are two lines is called the discriminant.

Thus for a conic bundle over a surface, the discriminant would be the curve in that surface over which the fibers are reducible, rather than irreducible. However if that curve is empty then both statements are true. Indeed in the cited reference, minimal [complex] conic bundles are said to be those with empty discriminant curve. That paper studies real conic bundles for which the term minimal is said to apply to those with imaginary discriminant curve.

If instead of requiring an embedding inducing the structure of conic on the fibers, one makes a definition that the fibers are only abstractly isomorphic to conics (as in this reference), then i suppose you could blow up the source threefold along a curve meeting each fiber at most once, and change a curve of irreducible fibers into reducible ones. That would seem to be a "non minimal" object you would want to exclude?

A general reference on [conic and] quadric bundles is Beauville's paper "Varietes de Prym et Jacobians intermediaire", in Ann. Sci. de l'Ecole Normale Sup., (4) 10 (1977), no.3, p.309 ff.

The concept of relatively minimal variety cited elsewhere here, seems related to minimal model theory, and hence presumably to the sort of bad example I proposed. As to the definition of minimal in the paper first cited in this answer, it seems not to be a sort that can always be achieved by modifications. I.e. it seems the threefold could be minimal and yet the discriminant curve still has real points. I am far from expert on this.

Answer 2

This paper gives the following definition of a relatively minimal variety:a relatively minimal variety $X'$ over a base $Y$ is a projective variety with at most $\mathbb{Q}$-factorial terminal singularities which has an extremal ray contraction $\phi: X'\to Y$ of fiber type, i.e., $\dim Y<\dim X$.