Address: Department of Mathematics, IIT Gandhinagar, Near Village Palaj, Gandhinagar – 382355, India
E-mail: samrutiya@iitgn.ac.in
Abstract: In this note, we prove that the $F$-fundamental group scheme is a birational invariant for smooth projective varieties. We prove that the $F$-fundamental group scheme is naturally a quotient of the Nori fundamental group scheme. For elliptic curves, it turns out that the $F$-fundamental group scheme and the Nori fundamental group scheme coincide. We also consider an extension of the Nori fundamental group scheme in positive characteristic using semi-essentially finite vector bundles, and prove that in this way, we do not get a non-trivial extension of the Nori fundamental group scheme for elliptic curves, unlike in characteristic zero.
AMSclassification: primary 14L15; secondary 14F05.
Keywords: F-fundamental group scheme, Frobenius-finite Vector bundles.
DOI: 10.5817/AM2020-1-21