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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.

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What exactly do you want to do with the result? Can you work locally, for example, in order to do what you want?ç –  Mariano Suárez-Alvarez Oct 3 '11 at 22:05
    
@Mariano I'm trying to compute the exceptional divisor in order to compute the Segre class of the diagonal in the product of varieties, following Corollary 4.2.2 in Fulton's Intersection Theory. –  Michael Capps Oct 4 '11 at 13:45

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