Let R=k[x1,…,xn] be a polynomial ring over a prefect field of positive characteristic. Let I be an equi-dimensional ideal in R and let J be a generic link of I in S=R[uij]c×r. We describe the parameter test submodule of S/J in terms of the test ideal of the pair (R,I) when a reduction of I is a complete intersection or almost complete intersection. As an application, we deduce a criterion for when S/J has F-rational singularities in these cases. We also compare the F-pure threshold of (R,I) and (S,J).
L. Ma, J. Page, R. R.G., W. Taylor, W. Zhang, "F-singularities under generic linkage", Journal of Algebra, Volume 505, 2018, Pages 194-210, ISSN 0021-8693, https://doi.org/10.1016/j.jalgebra.2018.02.024.