Why is this variety given by the vanishing of a linear form?

Question

In this paper, in Proposition 4.2.6 on page 97, the author says that if $H$ is the vanishing locus of a biform of bidegree $(d_1,d_2)$, then $\phi(H)$ is defined by the vanishing of a linear form, where $\phi$ is the composition of a product of Veronese embeddings of degrees $d_1$ and $d_2$, with a Segre embedding. Why is this true? And does it remain true if either $d_1$ or $d_2$ is zero?

Answer 1

The Veronese embedding $\nu_d: \mathbf P^n \rightarrow \mathbf P^N$ of degree $d$ is defined by all sections of $O(d)$, so $\nu_d^* O(1) = O(d)$. That means that degree-$d$ hypersurfaces in the source, which are zero loci of sections of $O(d)$, get mapped to intersections of $\nu_d(\mathbf P^n)$ with zero loci of sections of $O(1)$ --- that is, linear forms.

Similarly the Segre embedding $\mathbf P^n \times \mathbf P^m \rightarrow \mathbf P^{nm+n+m}$ is defined by sections of $O(1,1)$. So hypersurfaces of bidegree $(1,1)$ get mapped to intersections of the image with zero loci of linear forms.

Putting those two together gives the claim.

Finally, if $d_1$ or $d_2$ is zero, the Veronese map is not an embedding: it maps projective space to a point. Then the Segre map will be an isomorphism, and so the statement is just what we already said about the Veronese map.