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Suppose that $X$ is a complete intersection of quadrics in $\mathbb P^n_{\mathbb C}$. Is there some straightforward procedure to resolve the singularities of $X$?

For example, can one stratify singularities $X_{\rm sing}\subset X$ by some strata $X_1\subset X_2\subset \cdots \subset X_k=X_{\rm sing}$ so that $X$ can be resolved in $k$ steps by blowing consequently proper transforms of $X_i$, starting with blowing up $X$ in $X_1$?

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    $\begingroup$ For every irreducible projective variety $Y$, there exists a Veronese re-embedding such that the isomorphic image of $Y$ is one irreducible component of a complete intersection $X$ of quadrics in $\mathbb{P}^n$, cf. Mumford, "Varieties defined by quadratic equations." Every resolution of $X$ will also resolve $Y$. Thus, the resolution problem for all complete intersections of quadrics is just as hard as the general resolution problem. $\endgroup$
    – Jason Starr
    Apr 4, 2017 at 14:52
  • $\begingroup$ Very nice Jason! I must confess my ignorance here: I wasn't aware of this :( $\endgroup$
    – diverietti
    Apr 5, 2017 at 9:56

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