If $[x:y]$ are coordinates of $\mathbb{P}^1$ and $[X:Y:Z]$ are coordinates of $\mathbb{P}^2$, what do the following varieties look like?
- $V(x^2X+y^2Y+xyZ)\subset \mathbb{P}^2\times \mathbb{P}^1$
- $V(x^2X^2+y^2Y^2+xyZ^2)\subset \mathbb{P}^2\times \mathbb{P}^1$
As of now, I don't quite know how to visualize these varieties. A detailed geometric picture would be especially welcome.
Let's look at your first variety $V(x^2X+y^2Y+xyZ)\subset \mathbb{P}^2\times \mathbb{P}^1$.
You can visualize it as a family of varieties in two ways.
a) A one parameter family of lines in the projective plane parametrized by points of the projective line: you fix some point $(x_0:y_0)\in \mathbb P^1$ and you get the line $V(x^2_0X+y^2_0Y+x_0y_0Z)\subset \mathbb{P}^2$
b) A two parameter family of double points parametrized by the points of the projective plane: you fix some point $(X_0:Y_0:Z_0)\in \mathbb P^2$ and you get the double point $V(X_0x^2+Y_0y^2+Z_0xy)\subset \mathbb{P}^1$.
Of course these double points are to be taken in the scheme-theoretic sense: if $Z_0^2-4X_0Y_0=0$, we get only one physical point on the line
[Remember high-school, the equation $ax^2+bx+c=0$ and the case of coalesceing roots $b^2-4ac=0$ ?]
The second variety should be analyzed in a similar way.
This is just the tip of an iceberg called "algebraic families of schemes", involving flatness, Hilbert polynomial,...
But it's nice to see specific examples like here, as an unsophisticated, hands-on initiation.