The F-pure threshold and squarefree initial ideal as invariants of linkage

The generic link of an unmixed, radical (even a generic complete intersection) ideal is radical (in fact, prime). We show that this cannot be strengthened in general: The squarefreeness of the initial ideal and F-purity are not preserved along generic links. However, for several important cases in liaison theory, including generic height three Gorenstein ideals and the maximal minors of a generic matrix, we show that the squarefreeness of the initial ideal, F-purity, and even the F-pure threshold are each preserved along the n-th generic link for every n>0 by finding a property of such ideals which propagates along generic links. We use this property to establish the F-regularity of their generic links. Finally, we study the F-pure thresholds of the generic residual intersections of a complete intersection ideal and answer a related question of Kim--Miller--Niu.