The relative dimension of blow-up and singularities

Question

Let $X$ be an integral affine scheme of finite type over $\mathbb{C}$, $\mathrm{dim}\,X=d$. Let $Y\subset X$ be an integral closed subscheme of codimension $n>0$. We blow up $X$ along $Y$ and get a scheme $Bl_Y X$. Let $m_X(Y)$ be the maximum dimension of a closed subscheme of $Bl_YX$ that is proper (as a $\mathbb{C}$-scheme).

• if we fix $X$, and vary $Y$, is it true that $m_X(Y)$ has a uniform upper bound? Since $X$ is of finite type, I would think that this should be true.
• if we fix only $d$, and let $X$ and $Y$ vary, is it true that $m_X(Y)$ has a uniform upper bound?
• if we fix only $n$, and let $X$ and $Y$ vary, is it true that $m_X(Y)$ has a uniform upper bound?
• the above question under the assumption that $X$ is smooth/under the assumption that $Y$ is smooth;
• more generally, what does the dimension of the fibers of blow-up say about the singularities of the locus that is being blown-up?

Answer 1

For the first four questions, are you just asking what is the maximal dimension of a fiber of the blow-up map : $p : \mathrm{Bl}_{Y}(X) \longrightarrow X$? If that is the case, then for $X$ and $Y$ integral, the maximum of dimension of a fiber is $\dim X -1$ and it is attained when you blow-up a point in $X$.

The last question can be made into an interesting question if you transform it slightly. The dimension of the fibers can be arbitrary between $0$ and $d-1$, whatever the type of singularities, so nothing's interesting there. On the other hand, you could ask what is the relation between the variation of dimensions of the fibers of the blow-up along $Y$ and the variation of singularities of $X$ along $Y$.

This topic was studied a lot in the 60's, mostly in connection with Hironaka resolution's of singularities. Namely, Hironoka says that $X$ is normally flat along $Y$ if the exceptionnal divisor of the blow-up of $X$ along $Y$ is flat over $Y$. In particular, all fibers of the blow-up have the same dimension over points of $Y$. In case $Y$ is smooth, normal flatness is a notion of equi-singularity in a very strong sense. You can have a look at this question. It implies that all points of $Y$ have the same multiplicity in $X$, but in general it implies much more : namely that all generic polars varieties with respect to $X$ which contain $Y$ have the same multiplicity along $Y$ (this is a hard Theorem of Teissier).

Let me restrict to a simple case where things can be easily can be computed : assume that $X$ is a divisor in a smooth variety. If $Y$ is a smooth subvariety of $X$, then $X$ is normally flat along $Y$ if and only if all points of $Y$ have the same multiplicity in $X$. This computation in the case of hypersurfaces in ambient smooth varieties has been first carried out by Hironaka himself, but I think that Lejeune-Jalabert and Teissier gave a simpler approach to this result.

Very recently, this notion of normal flatness has proved to be quite useful in the context of non-commutative resolutions of singularities.

EDIT (22/04/19) : The answer to question 3 is no. Indeed let $\mathcal{N}_n$ be the nilpotent cone for $\mathrm{SL}_n$. The springer resolution: $$ \mu : T^* \left(\mathrm{SL_n}/B \right) \longrightarrow \mathcal{N}_n$$ is birational (a resolution of singularities) and for any $x \in \mathcal{O}_{subreg}$ , we have $\dim \mu^{-1}(x) \geq 1$ (where $\mathcal{O}_{subreg}$ is the subregular orbit). Since $\mathrm{codim}(\mathcal{O}_{subreg} \subset \mathcal{N}_n) = 2$, we deduce that the Richardson map is the blow-up along a subvariety of codimension $2$.

Finally the fiber of $\mu$ over $0$ is the complete flag varieties of $\mathrm{SL}_n$, it has dimension $\frac{n(n-1)}{2}$.