Veronese Surface - Biregular

Biregular

The image of a variety under the Veronese map is again a variety, rather than simply a constructible set; furthermore, these are isomorphic in the sense that the inverse map exists and is regular – the Veronese map is biregular. More precisely, the images of open sets in the Zariski topology are again open.

Biregularity has a number of important consequences. Most significant is that the image of points in general position under the Veronese map are again in general position, as if the image satisfies some special condition then this may be pulled back to the original point. This shows that "passing through k points in general position" imposes k independent linear conditions on a variety.

This may be used to show that any projective variety is the intersection of a Veronese variety and a linear space, and thus that any projective variety is isomorphic to an intersection of quadrics.