Let $X\subset \mathbb P_k^N$ be a $n$-dimensional smooth projective variety ($n\geq 2$) and $\phi_l:X^l\dashrightarrow Gr(l,N+1)$ ($l\leq n+1$) be the natural rational map which associates to a general $l$-tuple the $l$ dimensional linear space they span.
For $l=2$ there is an explicit resolution of indeterminacy locus by blowing up the diagonal in $X\times X$.
So I was wondering how (or if it is known) to find an explicit resolution for $l=3$.
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1$\begingroup$ You might as well restrict to the case that $X$ equals $\mathbb{P}^N_k$. If you know an explicit resolution in that case, then you can restrict that resolution to $X$. Now, for an $r$-tuple of nonzero vectors in $\mathbb{C}^{n+1}$, i.e., for an $(N+1)\times r$ matrix whose columns are nonempty, you are asking how to regularize the rational transformation to the Grassmannian via the "column space". The classical approach is via "complete collineations": iteratively blowup the smallest diagonal, then the strict transform of the locus of tuples of collinear points, etc. $\endgroup$– Jason StarrFeb 14, 2016 at 14:48
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