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roy smith
roy smith
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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 The concept of relatively minimal variety cited elsewhere here, seems unrelated to conic bundle structures, and related instead 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.

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 unrelated to conic bundle structures, and related instead to minimal model theory.)

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.

Source Link
roy smith
roy smith
  • 12.4k
  • 3
  • 78
  • 73

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 unrelated to conic bundle structures, and related instead to minimal model theory.)