A smooth counterexample to Nori’s conjecture on the fundamental group scheme

Abstract

Abstract. We show that Nori's fundamental group scheme π(X, x) does not base change correctly under extension of the base field for certain smooth projective ordinary curves X of genus 2 defined over a field of characteristic 2.

Non-properness of the functor of F-trivial bundles

“…But Christian Pauly has given a counter-example to Nori's second conjecture [16]. He constructs a nonconstant family of stable, F -trivial vector bundles, which is not constant.…”

On the “Galois Closure” for Finite Morphisms

“…If B/B is a faithfully flat extension, functoriality yields a morphism π( S/ B) → π(S/B) × B B, but this is by no means an isomorphism: see [8], §3 for a counterexample with S an integral projective curve and B and B algebraically closed fields. A counterexample with S a smooth curve has been given by Pauly in [12]. Theorem 4.9 Let B be a Dedekind scheme, (S, b) and (X, x) flat pointed B-schemes admitting a fundamental group scheme.…”

Vector Bundles Trivialized by Proper Morphisms and the Fundamental Group Scheme, II

“…Let f : Y −→ X be the fourth power of the Frobenius morphism (so Y is isomorphic to X a scheme). Pauly [Pa07,Proposition 4.1] constructs a locally free coherent sheaf over X × S, where S is a positive dimensional k-scheme, such that for every s ∈ S(k), the vector bundle E|X × {s} is stable and f * (E|X ×{s}) is trivial. Furthermore, for two different points s, t ∈ S(k), the sheaves E|X × {s} and E|X × {t} are not isomorphic.…”

“…We assume that k is of positive characteristic, and let F : X −→ X be the absolute Frobenius morphism. Define S(X, r, t) = isomorphism classes of stable vector bundles of rank r on X, whose pull-back by F t is trivial (Here we refrain from using the terminology F -trivial, since there is a question of stability which is not constant in the literature [Pa07], [MS08].) In their study of base change for the local fundamental group scheme and these bundles, Mehta and Subramanian [MS08] showed the following.…”

“…In view of Lemma 10 and Proposition 11, we can use [Pa07] to give an example of a smooth curve X having two extraordinary features: (1) there exists a finite morphism Y −→ X such that G(Y /X) is not finite and (2) the universal torsor X for the fundamental group scheme Π EF (X, x 0 ) is not reduced. Indeed, let X be the smooth curve constructed in [Pa07, (3.1) and Proposition 4.1]: it is a smooth projective curve defined by a single explicit equation in P 2 k ; here k is any field of characteristic two.…”