  We show that the Specht ideal of a two-rowed partition is perfect over an
arbitrary field, provided that the characteristic is either zero or bounded
below by the size of the second row of the partition, and we show this lower
bound is tight. We also establish perfection and other properties of certain
variants of Specht ideals, and find a surprising connection to the weak
Lefschetz property. Our results in particular give a self-contained proof of
Cohen-Macaulayness of certain $h$-equals sets, a result previously obtained by
Etingof-Gorsky-Losev over the complex numbers using rational Cherednik
algebras.
