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Suppose the coordinates of $\Bbb P^3$ are $\{x_1,x_2,x_3,x_4\}$, and in the affine patch $x_1=1$ an ideal $I=\langle\, f_1 , f_2,f_3\rangle$ is given ($f_i$'s are homogeneous polynomials on $\Bbb P^3$). I want to blow up the corresponding variety (which is a finite set of points). Algebraically I think the answer is just $U_i\, f_j=U_j\, f_i$.

But I want to know what is the corresponding refinement of the fan of $\Bbb P^3$? Should I go the maximal cone which defines the patch $x_1=1$ (say $\sigma$) and use the Cartier data $\{m_{\sigma}\}$? (another problem is that, $\{m_{\sigma}\}$ usually defines an ideal with monomial generators, but I'm talking about a general polynomial)

However, even when I use the Cartier data, the result I get seems wrong, for example in many examples, the ray that I should add is not even inside the cone!

I would appreciate if anyone can help me.

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    It is only true that blowups correspond to refinements of the fan if the blow-up variety is torus-invariant, which is not the case for general $f_i$. – 2017-02-27
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    I agree, so suppose the ideal generated by them is actually torus invariant... – 2017-02-27
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    The only torus-invariant radical ideals are the ideals of the form $\langle x_{i_1},\ldots, x_{i_k}\rangle$, where the $x_i$'s are variables. – 2017-02-28
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    Thanks, Can you give a reference about toric invariant ideal sheaves? – 2017-03-02

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