Some extremal contractions between smooth varieties arising from projective geometry

Abstract

We construct explicit examples of elementary extremal contractions, both birational and of fiber type, from smooth projective $n$-dimensional varieties, with $n \geq 4$, 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 are usually 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 $\mathbf{P}^{m}$, with $3 \leq m \leq 5$. We get another series of examples by considering special cubic hypersurfaces through some surfaces in $\mathbf{P}^5$, or some 3-folds in $\mathbf{P}^7$ 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 of the paper.