{'fraction_non_alphanumeric': 0.025995125913891144, 'fraction_numerical': 0.008123476848090982, 'mean_word_length': 5.350515463917525, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Hasse principle and weak approximation for pencils of Severi-Brauer and similar varieties.

Classes of varieties satisfying this principle are known other than quadrics (e.g. SeveriBrauer varieties and more generally complete varieties which are homogeneous spaces under a connected linear algebraic group). However counterexamples to the Hasse principle are known already in the class of rational varieties (varieties which become birational to projective space over a finite extension of the ground field). In 1970, Manin ([Ma71], [Ma86]) showed how most known counterexamples could be accounted for in terms of an obstruction based on the Brauer group of varieties. Further work (see [CT92] for a survey) has shown that for some classes of rational varieties this obstruction, henceforth referred to äs the Brauer-Manin obstruction, is the only obstruction to the Hasse principle. A reasonable class to investigate in this respect is that of varieties which are geometrically unirational, and also that of varieties which are geometrically birational to Fano varieties (a Fano variety is a smooth projective variety whose anticanonical class is ample; it is an open question whether such a variety is geometrically unirational).