I want to compute the blowup of $\mathbb{P}^3$ along the ideal $(w^3 + x^3 + y^3 + z^3, w^4 + \alpha wxyz)$ for fixed $\alpha \in k$. I've been working with blowups for a couple weeks, but this seems to require the use of a projective bundle, which I'm less familiar with.
Can someone walk me through the solution? I have other problems of a similar kind with which I can check my understanding. Thanks in advance!