Twelve Rational curves on Enriques surfaces

Abstract Given $$d\in {\mathbb {N}}$$ d ? N , we prove that any polarized Enriques surface (over field k of characteristic $$p \ne 2$$ p ? 2 or with a smooth K3 cover) degree greater than $$12d^2$$ 12 contains at most 12 rational curves d . For $$d>2$$ > construct examples surfaces high contain exactly degree- curves.