Russian Mathematical Surveys, 2023, Volume 78, Issue 3, Pages 563–565 DOI: https://doi.org/10.4213/rm10116e (Mi rm10116)

DOI: https://doi.org/10.4213/rm10116e

Bibliographic databases:

1. Moduli of bundles on curves

For simplicity we work over the field of complex numbers. Let $C$ be a smooth projective curve of genus $g \geqslant 2$. In [1] it was shown that for a general $C$ its bounded derived category of coherent sheaves $D(C)$ embeds in the derived category $D(M)$, where $M$ is the moduli space of stable rank $2$ bundles on $C$ with fixed determinant of odd degree. This result was independently obtained in [2]. A general statement, which was seemingly shown in [3], is that there is a semiorthogonal decomposition

A key ingredient in [1] was an explicit geometric description of $M$ for hyperelliptic curves. Let $C$ be hyperelliptic of genus $g$. Choose a coordinate on $\mathbb{P}^1$ so that the branching points $p_i=(1:a_i)$, $i=1,\dots,2g+2$, of the hyperelliptic projection are not at infinity. With the curve $C$ one associates the net of quadrics generated by

2. Moduli of parabolic bundles on $\mathbb{P}^1$

It is natural to try to generalise the previous results to the case when $V$ is of dimension $d=2g+1$ for some $g>1$. Once again, consider the net of quadrics (2) for distinct $a_1,\dots,a_{2g+1}$. It turns out that the variety of subspaces of dimension $g-1$ in $V$ isotropic with respect to $q_0$ and $q_\infty$ is isomorphic to the moduli space $\mathcal{M}$ of stable quasiparabolic bundles on rank $2$ and degree $0$ on $\mathbb{P}^1$ with weights $1/2$ at the marked points $p_i=(1:a_i)$ (see [5]). Meanwhile, $\mathcal{M}$ is isomorphic to the moduli space of rank $2$ bundles on stacky $\mathbb{P}^1$ with $\mathbb{Z}/2\mathbb{Z}$ structure at the points $p_i$ (see [6]); denote the latter space by $\mathcal{C}$. The following conjecture generalises the decomposition (1).

From Theorem 4.9 in [7] it follows that the derived category $D(\widetilde{S^k\mathcal{C}})$ carries a full exceptional collection. Thus, the same must be true for $D(\mathcal{M})$.

3. Computing the rank of $K_0(\mathcal{M})$

As evidence in support of our conjecture, we compute the ranks of the Grothendieck groups of the left- and right-hand sides of (3). The derived category of $\widetilde{S^k\mathcal{C}}$ has a semiorthogonal decomposition indexed by subsets whose components are the derived categories of intersections $\bigcap_{i\in I}H_i=\mathbb{P}^{k-|I|}$, where the index $I$ runs over the subsets $I\subseteq\{1,\dots,2g+1\}$ and $H_i$ is the $i$th hypersurface (see [7], Theorem 4.9). We conclude that the rank of the Grothendieck group of the right-hand side of (3) equals

Next we use the interpretation of $\mathcal{M}$ as the moduli space of parabolic bundles. A series of varieties $\mathcal{M}_0, \mathcal{M}_1,\dots,\mathcal{M}_{g-1}$ was constructed in [8] such that $\mathcal{M}_0=\mathbb{P}^{2g-2}$, $\mathcal{M}_1$ is a blowup of $\mathcal{M}_0$ in $(2g+1)$ points, $\mathcal{M}_{g-1}\simeq\mathcal{M}$, and $\mathcal{M}_{i+1}$ can be obtained from $\mathcal{M}_i$ via an anti-flip: in $\mathcal{M}_i$ one must blow up

The author os grateful to A. G. Kuznetsov and P. Belmans for interesting conversations.

Bibliography
1.	A. Fonarev and A. Kuznetsov, J. Lond. Math. Soc. (2), 97:1 (2018), 24–46
2.	M. S. Narasimhan, J. Geom. Phys., 122 (2017), 53–58
3.	J. Tevelev, Braid and phantom, 2023, 39 pp., arXiv: 2304.01825
4.	U. V. Desale and S. Ramanan, Invent. Math., 38:2 (1976), 161–185
5.	C. Casagrande, Math. Z., 280:3-4 (2015), 981–988
6.	I. Biswas, Duke Math. J., 88:2 (1997), 305–325
7.	D. Bergh, V. A. Lunts, and O. M. Schnürer, Selecta Math. (N. S.), 22:4 (2016), 2535–2568
8.	S. Bauer, Math. Ann., 290:3 (1991), 509–526

Citation in format AMSBIB

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\by A.~V.~Fonarev
\paper Derived category of moduli of parabolic bundles on $\mathbb{P}^1$
\jour Russian Math. Surveys
\yr 2023
\vol 78
\issue 3
\pages 563--565
\mathnet{http://mi.mathnet.ru/eng/rm10116}
\crossref{https://doi.org/10.4213/rm10116e}
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