39 research outputs found
Some extremal contractions between smooth varieties arising from projective geometry
We construct explicit examples of elementary extremal contractions, both
birational and of fiber type, from smooth projective n-dimensional varieties,
n\geq 4, onto smooth projective varieties, arising from classical projective
geometry and defined over sufficiently small fields, not necessarily
algebraically closed.
The examples considered come from particular special homaloidal and
subhomaloidal linear systems, which usually are degenerations of general
phenomena classically investigated by Bordiga, Severi, Todd, Room, Fano, Semple
and Tyrrell and more recently by Ein and Shepherd-Barron.
The first series of examples is associated to particular codimension 2
determinantal smooth subvarieties of P^m, 3\leq m\leq 5. We get another series
of examples by considering special cubic hypersurfaces through some surfaces in
P^5, or some 3-folds in P^7 having one apparent double point. The last examples
come from an intriguing birational elementary extremal contraction in dimension
6, studied by Semple and Tyrrell and fully described in the last section.Comment: 29 pages. Proc. London Math. Society, to appea
PGL(2) actions on Grassmannians and projective construction of rational curves with given restricted tangent bundle
We give an explicit parametrization of the Hilbert schemes of rational curves
C in P^n having a given splitting type of the restricted tangent bundle from
P^n to C. The adopted technique uses the description of such curves as
projections of a rational normal curve from a suitable linear vertex and a
classification of those vertices that correspond to the required splitting type
of the restricted tangent bundle. This classification involves the study of a
suitable PGL(2) action on the relevant Grassmannian variety