Fundamental groups of complements of wobbly divisors in intersections of two quadrics

@misc{bhattacharya2026fundamental,
  title = {{Fundamental groups of complements of wobbly divisors in intersections of two quadrics}},
  author = {Shomrik Bhattacharya and Yuki Matsubara},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.31421},
  eprint = {2609.31421},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.31421},
}

TY  - UNPB
TI  - Fundamental groups of complements of wobbly divisors in intersections of two quadrics
AU  - Shomrik Bhattacharya
AU  - Yuki Matsubara
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.31421
UR  - https://arxiv.org/abs/2609.31421
AB  - A smooth complete intersection X⊂ℙ^n+2_ℂ of two quadrics admits an interpretation as a moduli space of bundles. For n=2, it parametrizes stable rank $2$ parabolic bundles on ℙ^1 with five marked points, while for n=3 it parametrizes stable rank $2$ bundles of odd degree with fixed determinant on a genus $2$ curve. For general n, it can be interpreted as a moduli space of semistable twisted Spin-bundles. We study X∖ W, where W is the wobbly locus, consisting of bundles that admit a nonzero nilpotent Higgs field. We prove that W is an irreducible divisor for n≥ 3 and compute H_1(X∖ W,ℤ). Using a root-stack description, we identify π_1(X∖ W) with the kernel of a monodromy homomorphism from an orbifold mixed braid group. The Reidemeister-Schreier method then yields an explicit presentation in every dimension. For n=2, where X is a degree $4$ del Pezzo surface and W is the union of its sixteen (−1)-curves, we obtain a presentation with $10$ generators and $25$ relators and prove that both numbers are minimal among all finite presentations. For n=3, we show that π_1(X∖ W) admits a presentation with $4$ generators and $78$ relators.
ER  -