Минеев Дмитрий Александрович
Singular Complete Intersections of Quadrics and Cubics
A general smooth complete three-dimensional intersection of a quadric and a cubic is known to be non-rational. It is expected that any smooth variety of this type is non-rational. We consider intersections of quadrics and cubics containing a Veronese surface. They turn out to be singular but the singularities are isolated in general. We associate the conic duality construction to such an intersection which birationally transforms it into cubic hypersurface. This allows to study the rationality of the initial variety. We also study the singularities of such an intersection and extend the constructed birational transformation to Sarkisov link.