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Let $f$ be polynomial on a vector space $V$. Let $Z$ be the zero set of $f$ in $V$. Let $Z_{sing}$ be the singular part of $Z$. By Hironaka's desingularization theorem, there exists a birational map $\pi: X\to V$ such that the restriction map $\pi: X\backslash \pi^{-1}(Z_{sing})\to V\backslash Z_{sing}$ is an isomorphism; moreover $\pi^{-1}(Z)$ is the union of simple normal crossing divisors. For each divisor $E$, one can attach a pair of number $(N_E,n_E)$, where $N_E$ is the order of the function $\pi^*f$ along $E$ and $n_E$ is the order of $\pi^* \omega_V$ along $E$, where $\omega_V$ is the standard top form of $V$. These data are very important for the understanding of the singularity of $f$ and also have many applications in other fields.

I understand it is difficult to give explicit construction of resolution of singularity for each $f$. My question is, in the following examples, is there any references of the explicit desingularization and also the data $(N_E,n_E)$?

$(1)$. $f$ is the determinant on matrix space

$(2)$. $f=\sum_{i=1}^{n}x_iy_i$.

$(3)$. More generally, $f(A,B)=det(AB)$, where $A\in M_{nk}$ and $B\in M_{kn}$, $k\geq n$.

Thanks.

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For generic determinantal varieties, one log resolution is what is classically known as the "space of complete collineations". In the following article, Vainsencher computes the integers you want (although he uses slightly different notation than is common now).

MR0738261 (85f:14053) Reviewed
Vainsencher, Israel(BR-FPN)
Complete collineations and blowing up determinantal ideals.
Math. Ann. 267 (1984), no. 3, 417–432.
14M12 (14N10)

Edit. Also, before I was aware of Vainsencher's article (which I learned of thanks to Michael Thaddeus), I also computed these discrepancies in my preprint on Kodaira dimensions of the Kontsevich moduli spaces of genus $0$ stable maps to Fano hypersurfaces. So that is another source. Surprisingly, generic determinantal varieties turn out to be good models for the singularities that arise on Kontsevich moduli spaces, at least for "low" degree / homology class of the stable map (and one can use this to prove some Kontsevich spaces are terminal / canonical / log terminal / log canonical).

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    $\begingroup$ I'm actually not familiar with the concrete computation of these discrepancy data. Could you give some very basic reference for the computation? I'd like to do some very baby example computation, like the quadratic polynomial. Thank you. $\endgroup$
    – JJH
    Dec 4 '13 at 3:36

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