We give a geometric proof that the Hasse principle holds for the following varieties defined over global function fields: smooth quadric hypersurfaces, smooth cubic hypersurfaces of dimension at least in characteristic at least , and smooth complete intersections of two quadrics, which are of dimension at least , in odd characteristics.
"Hasse principle for three classes of varieties over global function fields." Duke Math. J. 166 (17) 3349 - 3424, 15 November 2017. https://doi.org/10.1215/00127094-2017-0034