Schmidt rank and singularities - Kazhdan, David; Lampert, Amichai; Polishchuk, Alexander (Ukrains’kyi Matematychnyi Zhurnal, 2023)
UDC 512.5 We revisit Schmidt’s theorem connecting the Schmidt rank of a tensor with the codimension of a certain variety and adapt the proof to the case of arbitrary characteristic. We also find a sharper result of this kind for homogeneous polynomials, assuming the characteristic does not divide the degree. Further, we use this to relate the Schmidt rank of a homogeneous polynomial (resp., a collection of homogeneous polynomials of the same degree) with the codimension of the singular locus of the corresponding hypersurface (resp., intersection of hypersurfaces). This gives an effective version of Ananyan–Hochster’s theorem [J. Amer. Math. Soc., 33, No. 1, 291–309 (2020), Theorem A].