Projective duality is a duality that associates to a (smooth) subvariety X of $\mathbb{P}^n$ the dual variety $X^*\subset\mathbb{P}^{n*}$ of tangent hyperplanes.

What makes the duality interesting is that if we work over an algebraically closed field of characteristic zero it is an actual duality, i.e. $(X^*)^*=X$.

Are there are references that analyse projective duality over not algebraically closed base fields or even dedekind schemes?

Thank you in advance for your help.