Noncommutative resolution of $SU_C(2)$

TL;DR

This work constructs a noncommutative resolution $\mathcal{D}$ of the singular moduli space $SU_C(2)$ for a genus $g\ge2$ curve, providing a semiorthogonal decomposition into blocks equivalent to $D^b(\mathrm{Sym}^{2k}C)$ for $2k\le g-1$, with multiplicities four (and an extra top block when $g$ is odd). The construction uses the symmetric-stack framework of Padurariu–Spenko–Van den Bergh and embeds the blocks into the Thaddeus moduli space $M$ via Fourier–Mukai kernels built from the universal bundle; in even genus, $\mathcal{D}$ is strongly crepant and categorifies the intersection cohomology of $SU_C(2)$, supporting conjectures about the rationality of $SU_C(2)$. A parallel analysis along the Hecke correspondence introduces quasi-BPS categories $\mathbb{B}_w$, and a Hecke Braid mutates two complementary semiorthogonal decompositions to relate the odd- and even-determinant pictures. The paper thereby advances a unified geometric and noncommutative perspective on $SU_C(2)$, its singularities, and birational behavior, while illuminating explicit mutation patterns via weaving analogies such as Cross Warp and Farey Twill.

Abstract

We study the derived category of the moduli space $SU_C(2)$ of rank $2$ vector bundles on a smooth projective curve $C$ of genus $g\ge 2$ with trivial determinant. This generalizes the recent work by Tevelev and Torres on the case with fixed odd determinant. Since $SU_C(2)$ is singular, we work with its resolution of singularities, specifically with the noncommutative resolution constructed by Pădurariu and Špenko--Van den Bergh (in the more general setting of symmetric stacks). We show that this noncommutative resolution admits a semiorthogonal decomposition into derived categories of symmetric powers $Sym^{2k}C$ for $2k\le g-1$. In the case of even genus, each block appears four times. This is also true in the case of odd genus, except that the top symmetric power $Sym^{g-1}C$ appears twice. In the case of even genus, the noncommutative resolution is strongly crepant in the sense of Kuznetsov and categorifies the intersection cohomology of $SU_C(2)$. Since all of its components are "geometric," our semiorthogonal decomposition provides evidence for the expectation, which dates back to the work of Newstead and Tyurin, that $SU_C(2)$ is a rational variety. Finally, we study mutations of semiorthogonal decompositions on the Hecke correspondence, answering a question of Pădurariu and Toda.

Figures (5)

• Figure 1: The basic Cross Warp mutation, cf. tevelevbraid*Figure 7.
• Figure 2: Stacking the crosswarp mutation (with $m_1=0$). See also tevelevbraid*Figure 8.
• Figure 3: Modified Plain Weave in genus $5$, cf. tevelevbraid*Figure 13.
• Figure 4: The inductive step.
• Figure 5: The mutation of \ref{['thm:crosswarp_mutation']} in two steps, cf. tevelevbraid*Figure 9.

Theorems & Definitions (103)

• Theorem 1.1
• Conjecture 1.2: Two-Ray Game Conjecture (1)
• Conjecture 1.3: Two-Ray Game Conjecture (2)
• Theorem 2.2
• Remark 2.3
• Remark 2.4
• Lemma 2.6: cf. tevelevbraid*Corollary 2.10
• Proposition 2.7: tevelevtorresbgmn*Proposition 3.18
• Remark 2.8
• proof : Proof of \ref{['lem:ft_main']}