Finite Pullbacks, Frobenius Periodicity, and Finite Trivializations

Time: 14:30–16:30, 11 September, 2026

Venue: Room 612, Building A6

Online Zoom link:  https://zoom.us/j/99636681387?pwd=0WscBnehOJig68SqctGluVuA3RwraE.1

Abstract: The final talk studies how numerical flatness and strong semistability behave under finite covers. We first prove that numerical flatness is both preserved and detected by finite surjective pullback; consequently, on curves, the same is true for strong semistability in degree zero. We then turn to Frobenius-periodic bundles and the theorem of Lange–Stuhler relating Frobenius periodicity to trivialization by a finite étale cover. Particular attention will be paid to the finite-field hypothesis and to what remains valid over arbitrary fields of positive characteristic. The seminar concludes with an explicit rank-two example: an extension of $(\mathcal O_X)$ by itself whose class is fixed by Frobenius. Using the Artin–Schreier sequence, we construct a finite étale cover on which the extension becomes trivial.