I'm looking for a easy to work with definition of a blowup with an example/someone to check my undestanding of a blowup.
I understand that blowups are a method that we use to deal with singularites of plane curves.
The most famous example is using a blow up to deal with $y^2 = x^3 + x^2$
So with a blow up, we want to blow up a point in affine space $A^n$.
In the example, we have the affine space $A^2$ and we want to blow up a point $(x_1,x_2)$ in this case $(0,0)$
Then we look at the variety space $A^n \times \mathbb{P}^{n-1}, A^n = (x_1,...,x_n),\mathbb{P}^{n-1}=(X_1,...,X_{n-1})$
We take the subset $B = \{A^n \times P^{n-1} |x_iX_j=y_jX_i\}$
I think we call this set of points the blowup.
So in terms of our example,
we introduce $y=tx$ (what is this called)?
Then $y^2=x^3 + x^2 \implies t^2x^2 = x^3 + x^2 \Leftrightarrow x^2(t^2 - x - 1) =0$
From here we get $x^2 = 0 \Leftrightarrow x=0$ and $y=0$
Also $t^2-x-1=0 \Leftrightarrow t^2 = x+1$
Now this curve $t^2=x+1$ is the resolved curve of $y^2 = x^3 + x^2$ and we have the exceptional curve $x=0$
and this is the blowup of the point (0,0)