Variant of Segre embedding

Question

We work over a field $k$. We know that there is the Segre embedding $\def\P{\mathbb{P}} \P^2 \times \P^1 \to \P^5$. Now I want an embedding of $\P^2 \times \def\A{\mathbb{A}}\A^1$ into some projective space. (Or into some affine space, which is however impossible.)

Of course, the Segre embedding induces an embedding $\def\P{\mathbb{P}} \P^2 \times \A^1 \to \P^5$. However, I want the embedding to be as "simple" (in particular low-dimensional) as possible, so I was wondering if there is for example an embedding $\P^2 \times \A^1 \to \P^4$ ?

Background: I want to work with the space $\mathbb{P}^2 \times \mathbb{A}$ in the computer algebra system SageMath. However, in SageMath the only allowed ambient spaces are projective space and affine space. If you know another computer algebra system in which I could work directly with $\mathbb{P}^2 \times \mathbb{A}$, that would be very helpful as well.

Answer 1

Maybe using toric varieties ? You can embed in $\mathbb{P}^2 \times \mathbb{P}^1$.

sage: A1 = toric_varieties.A1()
sage: P2 = toric_varieties.P2()
sage: V = P2.cartesian_product(A1); V
3-d toric variety covered by 3 affine patches