minimality/universality of the Springer resolution of a determinantal variety

Question

Let $X\subset P^n$ be a singular determinantal variety and $S\to X$ its Springer resolution. Let $X'\to X$ another resolution of singularities (say, a blow-up). Does $S$ have some minimality/universality property? I.e.: does $X'\to X$ always factor through $S\to X$?

Answer 1

I am not sure what you call a "Springer resolution", because I guess most determinantal varieties do not have any Springer resolution.

But anyway, say that we look at $X = \{ A \in End(V)\, \text{such that}\, A^2 = 0\, \text{and}\, \operatorname{rk}(A) \leq 1 \}$, then it has two Springer resolutions which are given by the cotangent bundles over $\mathbb{P}(V)$ and $\mathbb{P}(V^*)$.

Now, of course these two resolutions are isomorphic, but that it somehow a coincidence. If insead of looking at $X$, I look at $X_Y = \{ A \in End(E)\, \text{such that}\, A^2 = 0\, \text{and}\, \operatorname{rk}(A) \leq 1 \}$, where $E$ is the total space of rank $\geq 3$ vector bundle over some variety $Y$, then I still have two Springer resolutions of $X_Y$ an this time they are really non-isomorphic (unless $E \simeq E^*$).

In fact you can even prove that there is no well-defined morphism from one of the resolution to the other. Indeed, both resolutions are related to each other by a flop, so if you had a well defined morphism between the two (making all possible diagrams commutative), then they would be isomorphic.

However, there is a nice conjecture of Bondal and Orlov which says that these two resolutions should be derived equivalent. More precisely, you have the following:

Conjecture(Bondal-Orlov) Let $X_1 \rightarrow X \leftarrow X_2$ be two crepant resolutions of a Gorenstein rational singularity, then $X_1$ and $X_2$ are derived equivalent.

The flop case is a special case of this conjecture and relates for instance different Springer resolutions of a nilpotent orbit.

The conjecture of Bondal and Orlov is even stronger in its original form and suggests the following (which relates to the "minimality" you expect from the Springer resolutions):

Conjecture(Bondal-Orlov) Let $X_0 \rightarrow X$ be a crepant resolution of a Gorenstein rational singularity. Then, for any other resolution of singularities $X_1 \rightarrow X$, there is a fully faithful functor: $$ D^b(X_0) \hookrightarrow D^b(X_1).$$

However, this last conjecture should be considered not too naively. Unless we are in some very specific situations (say flips or flops), there is absolutely no way to construct a "good" natural functor from $D^b(X_0)$ to $D^b(X_1)$. This a conjecture may be just seen as a interesting clue that derived categories might be a nice tool if you want to define, express or test minimality of a resolution of singularities.