The tame fundamental group schemes of curves in positive characteristic
Shusuke Otabe
TL;DR
The paper investigates tame fundamental group schemes $\pi^{\mathrm{tame}}(U)$ for smooth curves over algebraically closed fields of characteristic $p>0$, extending Grothendieck–Tamagawa specialization theory to a cospecialization framework for finite quotients in the class of elementary linearly reductive group schemes $\mathscr{D}$. It proves an injectivity result for the cospecialization map and shows compatibility with base change, enabling reconstruction of numerical invariants $(g,n,\gamma)$ from tame data; it also establishes that tame fundamental group schemes vary with the curve and are not constant in families. The work unifies the Nori fundamental gerbe, root stacks, and finite linearly reductive torsors to build a cohesive theory of tamely finite group schemes in positive characteristic, and provides a framework for understanding how curve invariants are encoded in tame fundamental group information. These results advance anabelian-type questions in positive characteristic by linking group-scheme data with geometric invariants and demonstrating non-constancy phenomena on moduli spaces.
Abstract
The tame fundamental group scheme for an algebraic variety is the maximal linearly reductive quotient of Nori's fundamental group scheme. In this paper, we study the tame fundamental group schemes of smooth curves defined over algebraically closed fields of positive characteristic and develop the theory of cospecialization maps for them. As a result, we see that the tame fundamental group schemes heavily depend on the curves. We also see that numerical invariants of curves can be reconstructed from the tame fundamental group schemes.