I am studying *analytic spreads* from *Bruns-Herzog's* book. The definition is clear but calculation of the analytic spread of an ideal is hard for me in practice. I wonder if it is hard for you too. 



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> Is there is a intuitive interpretation of an analytic spread that can help for a better understanding of it? 
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> Is there any software that can help to calculate the analytic spread of (perhaps some special) ideals?
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intuitive interpretation of analytic spread
Question 1: Interpretation
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Ok, so here is how I like to think of it from a geometric perspective.


**Definition:** (Under moderate hypotheses) *analytic spread* of an ideal $J$ is the minimum number of generators of an ideal with the same integral closure as $J$.


**Definition:** Two ideals $J, J'$ have *same integral integral closure* if and only if their normalized blowups $X = X'$ are the same with equal pullbacks $J \cdot O\_X = J' \cdot O\_X$. 


The number of generators of an ideal of course gives a bound on the number of affine charts you need to cover a blowup.


Putting these together you obtain that:


**Fact:** Analytic spread of $J$ is a canonical upper bound on the number of affine charts you need to cover the blowup of $J$. In other words, it measures the complexity of a blowup of $J$.


Let's look at the usual:


**Example:** Consider $J = \langle x^3, x^2y, xy^2, y^3 \rangle \subseteq k[x,y]$. Then the analytic spread is two, corresponding to the ideal $J' = \langle x^3, y^3 \rangle$. The normalized blowup of $J'$ is the same as the blowup of $J$, which is just blowing up the origin in $\mathbb{A}^2$. On the other hand, you only need two charts to cover the blowup of $J$, the chart corresponding to inverting $x^3$, and the chart corresponding to inverting $y^3$.


Now, it could be that the analytic spread is *exactly* the minimum number of charts needed to cover the blowup, I don't know (if it's important I could think of a couple people I would ask).


Finally, if you were looking for more algebraic interpretations, I would see if anything in the Swanson-Huneke book on integral closure helps.


Question 2: Computing it
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Did you try:


<http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.6/share/doc/Macaulay2/ReesAlgebra/html/_analytic__Spread.html>


