# Seshadri's desingularisation in the Hodge diamond cutter, and a bold proposal

The functionality outlined below, and much more, is implemented in **Hodge diamond cutter**, which can be used in Sage.
If you use it for your research, please cite it using .

As I was cleaning up things I ran into my print-out of del Baño: On the motive of moduli spaces of rank two vector bundles over a curve, which I apparently printed on December 15, 2017. This marks the start of my interest in these moduli spaces (which have an extremely rich and beautiful history!) and I noticed I never really looked at Section 3. In this section the Hodge numbers for Seshadri's desingularisation of $\mathrm{M}_C(2,\mathcal{O}_C)$ are computed, and I couldn't resist having a little fun with this.

This is now implemented in the Hodge diamond cutter. Mind that there are 2 minor typos in the formula in the paper, which one can quickly fix by looking at the statements of the results from which the corollary follows.

One thing which I'd find interesting is to figure out what the derived category of this variety looks like. Or maybe one should rather look for the structure of a categorical crepant resolution.

If we do it for Seshadri's resolution and $g=3$, it seems we get $4$ exceptional objects (which is reasonable from the index being 4), 2 copies of $\mathbf{D}^{\mathrm{b}}(\operatorname{Sym}^2C)$ and then something which can only be categorical it seems, with Hochschild homology in degrees $-2,0,2$ of dimension $6,84,6$. But no copy of $\mathbf{D}^{\mathrm{b}}(C)$. Maybe a better candidate can be found though! And I should probably look at what is known about intersection cohomology.

Added a little while after publishing this: In case you wish to play around with the intersection cohomology (which somehow should measure the size of a noncommutative crepant resolution), you can use

~~Still, the picture isn't obvious to me, but at least copies of $\mathbf{D}^{\mathrm{b}}(C)$ seem to be naturally present now.~~ The formula is taken from this paper by the way.

### A proposal

A little more playing gives a clear picture: the noncommutative crepant resolution $\mathcal{C}$ of $\mathrm{M}_C(2,\mathcal{O}_C)$ looks like it might have a semiorthogonal decomposition of the form \begin{equation} \mathcal{C} = \begin{cases} \langle\mathbf{D}^{\mathrm{b}}(\mathrm{pt}), \mathbf{D}^{\mathrm{b}}(\mathrm{pt}), \mathbf{D}^{\mathrm{b}}(\mathrm{pt}), \mathbf{D}^{\mathrm{b}}(\mathrm{pt}),\\ \quad\mathbf{D}^{\mathrm{b}}(\operatorname{Sym}^2C), \mathbf{D}^{\mathrm{b}}(\operatorname{Sym}^2C), \mathbf{D}^{\mathrm{b}}(\operatorname{Sym}^2C), \mathbf{D}^{\mathrm{b}}(\operatorname{Sym}^2C),\\ \quad\ldots,\\ \quad\mathbf{D}^{\mathrm{b}}(\operatorname{Sym}^{g-2}C), \mathbf{D}^{\mathrm{b}}(\operatorname{Sym}^{g-2}C), \mathbf{D}^{\mathrm{b}}(\operatorname{Sym}^{g-2}C), \mathbf{D}^{\mathrm{b}}(\operatorname{Sym}^{g-2}C)\rangle & g\equiv 0\bmod 2 \\ \langle\mathbf{D}^{\mathrm{b}}(C), \mathbf{D}^{\mathrm{b}}(C), \mathbf{D}^{\mathrm{b}}(C), \mathbf{D}^{\mathrm{b}}(C), \\ \quad\mathbf{D}^{\mathrm{b}}(\operatorname{Sym}^3C), \mathbf{D}^{\mathrm{b}}(\operatorname{Sym}^3C), \mathbf{D}^{\mathrm{b}}(\operatorname{Sym}^3C), \mathbf{D}^{\mathrm{b}}(\operatorname{Sym}^3C), \\ \quad\ldots,\\ \quad\mathbf{D}^{\mathrm{b}}(\operatorname{Sym}^{g-2}C), \mathbf{D}^{\mathrm{b}}(\operatorname{Sym}^{g-2}C), \mathbf{D}^{\mathrm{b}}(\operatorname{Sym}^{g-2}C), \mathbf{D}^{\mathrm{b}}(\operatorname{Sym}^{g-2}C)\rangle & g\equiv 1\bmod 2 \\ \end{cases} \end{equation}

So for $g$ even there are 4 copies of the derived category of even symmetric powers of $C$, for the powers $0,2,\ldots,g-2$, whilst for $g$ odd there are 4 copies of the derived category of odd symmetric powers of $C$, for the powers $1,3,\ldots,g-2$.

For $g=2$ this is good, because we're considering a noncommutative crepant resolution of $\mathbb{P}^3$, which is just $\mathbb{P}^3$ itself and there are 4 exceptional objects. For $g=3$ the numbers suggest a semiorthogonal decomposition with just 4 copies of $\mathbf{D}^{\mathrm{b}}(C)$. Can that be true?!

We also see the index of the (singular) moduli space being 4 in this picture.

What is odd is that for $g$ odd there are no exceptional objects in the decomposition. I somehow expect there to be some of them, as we are considering a conjectural noncommutative crepant resolution of a Fano variety, and for smooth Fanos we always have the structure sheaf as an exceptional object.

So this is just some wild speculation, based on the following code:

The idea is that the columns of the "intersection Hodge diamond" correspond to the Hochschild homology of a categorical crepant resolution. I don't know whether that makes sense though, it's just Sunday afternoon speculation.

I should create documentation using Sphinx I guess. That should making the code easier I hope.