The integral K0-ring of the stack of smooth hyperelliptic curves
Abstract
We compute the integral K0-ring of the moduli stack ℋg of smooth hyperelliptic curves of genus g≥2 over a field of characteristic zero. We show that it is a free abelian group of rank 8g+2 for even g and 8g+4 for odd g. We use the isotypic decomposition of the K0 of a related μg+1-gerbe to obtain presentations for both even and odd g's from the standard quotient presentation of ℋg. Thus, our technique differs from the CH•(ℋg) computations of Edidin--Fulghesu, Fulghesu--Viviani, Di Lorenzo, and the K0(ℳ2) computation of Edidin--Hu. Using Pagani's description of the inertia stack of ℋg, we apply Riemann--Roch to interpret the rank of K0(ℋg) as the number of inertia components. Finally, we compare the K0 and Chow rings of ℋg through the gamma-filtration and give presentations to the associated graded rings grγK0(ℋg)⊗𝔽p for every prime p.
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BibTeX
@misc{zeng2026integral,
title = {{The integral $K_0$-ring of the stack of smooth hyperelliptic curves}},
author = {Michael Ruofan Zeng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27267},
eprint = {2610.27267},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27267},
}
RIS
TY - UNPB
TI - The integral K_0-ring of the stack of smooth hyperelliptic curves
AU - Michael Ruofan Zeng
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.27267
UR - https://arxiv.org/abs/2610.27267
AB - We compute the integral K_0-ring of the moduli stack ℋ_g of smooth hyperelliptic curves of genus g≥2 over a field of characteristic zero. We show that it is a free abelian group of rank 8g+2 for even g and 8g+4 for odd g. We use the isotypic decomposition of the K_0 of a related μ_g+1-gerbe to obtain presentations for both even and odd g's from the standard quotient presentation of ℋ_g. Thus, our technique differs from the CH^•(ℋ_g) computations of Edidin--Fulghesu, Fulghesu--Viviani, Di Lorenzo, and the K_0(ℳ_2) computation of Edidin--Hu. Using Pagani's description of the inertia stack of ℋ_g, we apply Riemann--Roch to interpret the rank of K_0(ℋ_g) as the number of inertia components. Finally, we compare the K_0 and Chow rings of ℋ_g through the gamma-filtration and give presentations to the associated graded rings gr_γK_0(ℋ_g)⊗𝔽_p for every prime p.
ER -