$F$-purity and the $F$-pure threshold as invariants of linkage

The generic link of an unmixed radical ideal is radical (in fact, prime). We show that the squarefreeness of the initial ideal and $F$-purity are, however, not preserved along generic links. On the flip side, 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 the $F$-pure threshold are each preserved along generic links by identifying a property of such ideals which propagates along generic links. We use this property to establish the $F$-regularity of the generic links of such ideals. Finally, we study the $F$-pure threshold of the generic residual intersections of a complete intersection ideal and answer a related question of Kim–Miller–Niu.