On base change of the fundamental group scheme (Q2249630)

In [Proc. Indian Acad. Sci., Math. Sci. 91, 73--122 (1982; Zbl 0586.14006)] \textit{M. V. Nori} introduced the so-called fundamental group scheme of a proper connected reduced scheme \(X\) over a field \(k\) as the affine group scheme associated to the (neutral) Tannakian category of essentially finite vector bundles on \(X\) with a fixed fibre functor \(x\). He also conjectured that if \(X\) is a complete geometrically connected and reduced scheme over an algebraically closed field \(k\) and \(k\subseteq K\) is an arbitrary extension of algebraically closed fields then the natural map \[ h_X: \pi_1(X\times_k \text{Spec}(K), x) \to \pi_1(X,x)\times_k \text{Spec}(K) \] is an isomorphism of affine group schemes over \(K\).'' However, this conjecture is false in general in any positive characteristic. Firstly, it has been disproven for curves with cuspidal singularities by \textit{V. B. Mehta} and \textit{S. Subramanian} [Invent. Math. 148, No. 1, 143--150 (2002; Zbl 1020.14006)]. More recently, \textit{C. Pauly} [Proc. Am. Math. Soc. 135, No. 9, 2707--2711 (2007; Zbl 1115.14026)] has given an example of a smooth curve in characteristic 2 where the conjectured base change property fails. In the article under review the author is extending this result to smooth curves in any positive characteristic by providing explicit smooth examples where the base change property fails.