We introduce four invariants of algebraic varieties over imperfect fields, each of which measures either geometric non-normality or geometric non-reducedness. The first objective of this article is to establish fundamental properties of these invariants. We then apply our results to curves over imperfect fields. In particular, we establish a genus change formula and prove the boundedness of non-smooth regular curves of genus one. We also compute our invariants for some explicit examples.
"Invariants of algebraic varieties over imperfect fields." Tohoku Math. J. (2) 73 (4) 471 - 538, 2021. https://doi.org/10.2748/tmj.20200611