| Russian Mathematical Surveys |
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This article is cited in 1 scientific paper (total in 1 paper) Brief communications Derived category of moduli of parabolic bundles on A. V. Fonarevab a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow b National Research University Higher School of Economics
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     1. Moduli of bundles on curvesFor 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
$$
\begin{equation}
D(M) =\bigl\langle \mathcal{O}, \mathcal{O}(1), D(C), D(C)(1), \ldots, D(S^{g-2}C), D(S^{g-2}C)(1), D(S^{g-1}C) \bigr\rangle,
\end{equation}
\tag{1}
$$
where $S^iC$ denotes the $i$th symmetric power of the curve $C$. 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
$$
\begin{equation}
q_0=a_1x_1^2+a_2x_2^2+\cdots+a_{d}x_{d}^2, \qquad q_\infty=-(x_1^2+x_2^2+\cdots+x_{d}^2),
\end{equation}
\tag{2}
$$
where $d=2g+2$ and the $x_i$ for $i=1,\dots,d$ are coordinates on a fixed $d$-dimensional vector space $V$. In turns out that $M$ is isomorphic to the space of $(g-1)$-dimensional subspaces in $V$ which are isotropic with respect to $q_0$ and $q_\infty$ (see [4]).
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). Conjecture. There is a semiorthogonal decomposition where $\widetilde{S^k\mathcal{C}}$ denotes the root stack obtained from $\mathbb{P}^k$ by extracting the square root from $2g+1$ hyperplanes in general position.. 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
$$
\begin{equation*}
l_g=\displaystyle\sum_{k=0}^{g-1}\displaystyle\sum_{t=0}^k(t+1) \begin{pmatrix} 2g+1 \\ k-t\end{pmatrix}.
\end{equation*}
\notag
$$
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
$$
\begin{equation*}
n_i=\begin{pmatrix} 2g+1 \\ i+1 \end{pmatrix} + \begin{pmatrix} 2g+1 \\ i-1 \end{pmatrix} + \begin{pmatrix} 2g+1 \\ i-3 \end{pmatrix} + \cdots
\end{equation*}
\notag
$$
subvarieties isomorphic to $\mathbb{P}^i$, and then blow down the exceptional divisors to subvarieties isomorphic to $\mathbb{P}^{2g-3-i}$ in $\mathcal{M}_{i+1}$. The rank of $\operatorname{rk}K_0(\mathcal{M}_0)$ is $2g-1$, while the blowup formula implies that $\operatorname{rk} K_0(\mathcal{M}_{i+1})-\operatorname{rk} K_0(\mathcal{M}_i)=n_i(2g-3-2i)$. We can thus compute $r_g=\operatorname{rk}\mathcal{M}=\operatorname{rk}\mathcal{M}_{g-1}$. Collecting the coefficients at $ \begin{pmatrix} 2g+1 \\ i \end{pmatrix}$ we conclude that $l_g=r_g$. It turns out that both quantities can be expressed by a closed formula. The author os grateful to A. G. Kuznetsov and P. Belmans for interesting conversations.
Citation in format AMSBIB
\Bibitem{Fon23} |
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