I am working with Shafarevich's "Basic Algebraic Geometry 1".
Example 1.15: The map $f(t)=(t^2,t^3)$ is a regular map on the line $\mathbb{A}^1$ to the curve given by $y^2=x^3$.
I am not familiar with regular functions yet, so I have some problems to see how this works.
So $f:\mathbb{A}^1\to V(y^2-x^3)$. But why do we see this out of $f(t)=(t^2,t^3)$? Because not all $t\in\mathbb{A}^1$ would fulfill $y^2-x^3=0$ or how is it meant?
I hope you understand where my problem is.. I would be happy for any further explanation of this example! Thanks and best, Luca