When is the Blow-up of $\mathbb{P}^n$ dominant?

Question

Let $X\subset\mathbb{P}^n$ be some algebraic variety. Lets assume that there are some homogeneous polynomials $f_0,\ldots,f_n$, all of degree $d$, that generate the vanishing ideal of $X$. When is the rational map $\mathbb{P}^n\dashrightarrow\mathbb{P}^n, x\mapsto (f_0(x):\cdots:f_n)$ dominant? Are there some nice criteria? What are examples where it is not the case?

Answer 1

Here is an example. Let $A$ and $B$ be vector spaces of dimension 2 and 5. Let $$ X = \mathbb{P}(A) \times \mathbb{P}(B) \subset \mathbb{P}(A \otimes B). $$ The ideal of $X$ is generated by quadrics, which span the space $$ \mathrm{Ker}(S^2(A \otimes B)^\vee \to S^2A^\vee \otimes S^2B^\vee) = \wedge^2A \otimes \wedge^2B \cong \wedge^2B. $$ Note that $\dim(A \otimes B) = 10 = \dim(\wedge^2B)$. But the image of the corresponding map $$ f \colon \mathbb{P}(A \otimes B) \dashrightarrow \mathbb{P}(\wedge^2B) $$ is the Grassmannian $$ \mathrm{Gr}(2,B) \subset \mathbb{P}(\wedge^2B). $$