# Variant of Segre embedding

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.

Maybe using toric varieties ? You can embed in $$\mathbb{P}^2 \times \mathbb{P}^1$$.
sage: A1 = toric_varieties.A1()