Enriques surfaces with non-generic non-degeneracy

We study the non-degeneracy invariant $\mathrm{n}\mathrm{d}(Y)$ of complex Enriques surfaces in families. Our first main result shows that $\mathrm{n}\mathrm{d}(Y)$ cannot increase under specialization. The second main result is the conclusion of the computation of the non-degeneracy invariant for the $155$ families of $(\tau ,\bar{\tau })$-generic surfaces introduced by Brandhorst and Shimada. Of the previously known $144$ cases, only $3$ satisfy $\mathrm{n}\mathrm{d}(Y)\mathrm{\ne }10$, which is the non-degeneracy invariant of a general Enriques surface. The remaining $11$ families studied in this article also have non-generic non-degeneracy. To compute this, we produce upper bounds on $\mathrm{n}\mathrm{d}(Y)$ by refining this invariant into two others: the Fano and Mukai non-degeneracy invariants, which are related to two different classes of projective realizations of Enriques surfaces. As a result, we find the first known examples of Enriques surfaces with $\mathrm{n}\mathrm{d}(Y)=9$.