I am looking for information on blowing up along an ideal in a product of varieties. After extensive searching through several textbooks I cannot find an explicit method for doing so. Specifically, I am trying to blow up along the diagonal in the product of two projective varieties.
To clarify, I attempting to perform explicit computations using Macaulay2. I have projective variety $X$ that lives in $\mathbb P^{12}$, and I am sending the tensor product $X\times X$ to $\mathbb P^{168}$ using the Segre embedding and attempting to blow up the diagonal of $X\times X$ there. However, I am running into physical memory problems due to the massive number of polynomials needed to describe the ideal in $\mathbb P^{168}$. So what I'm really looking for is a method for blowing up along an ideal in a product of varieties that doesn't rely on the Segre embedding.