Golyshev's canonical strip hypothesis

As a Christmas treat almost 2 years ago I produced some colourful pictures of the roots of Hilbert polynomials of smooth toric Fano varieties. These were meant to illustrate Golyshev's hypothesis, which predicts that these roots are to lie in the "canonical strip". Jointly with Sergey Galkin and Swarnava Mukhopadhyay we have found counterexamples to this hypothesis in higher dimension, using non-toric Fano varieties: namely moduli of vector bundles of rank 2 and odd determinant, on curves of genus $g$).

The two players

There is an interesting parallel between

  • moduli of rank 2 bundles with odd determinant on a curve of genus $g\geq 2$
  • moduli of parabolic rank 2 bundles on $\mathbb{P}^1$ with weight 1/2 at $2g+1$ points, for $g\geq 2$

The first is a Fano variety of Picard rank 1, index 2 and dimension $3g-3$, whilst the second is a Fano variety of Picard rank $2g+2$, index 1, and dimension $2g-2$. and more importantly, for both we have a Verlinde formula expressing dimensions of global sections of the anticanonical divisor. Hence we can determine their Hilbert polynomials, and compute the location of their roots.

The pictures

Doing some totally unrelated reading on moduli of parabolic bundles a few days ago and coming across this Verlinde formula again, I wanted to (I might suffer from some mild ADD?)

  • check the possible failure of Golyshev's hypothesis for these moduli spaces of parabolic bundles
  • make a colourful version of the plot we made in our paper and the analogous plot for the parabolic case

In the two plots below, once the roots go outside the interval $[-2,0]$ (resp. $[-1,0])$, the hypothesis fails. This happens for $g\geq 10$ (resp. $g\geq 8$). So this gives a second class of counterexamples! And in fact, whilst the counterexample we originally constructed had dimension 27, we now already see a counterexample in dimension 14.

I think it would be interesting to understand the asymptotic behaviour which one observes in these pictures. Let me know if you have any suggestions!