A rank-$2$ vector bundle on ${\mathbb{P}}^2\times {\mathbb{P}}^2$ and projective geometry of nonclassical Enriques surfaces in characteristic 2

We construct a rank-$2$ indecomposable vector bundle on ${\mathbb{P}}^2\times {\mathbb{P}}^2$ in characteristic $2$ that does not come from a bundle on ${\mathbb{P}}^2$ by factor projection nor from a bundle on ${\mathbb{P}}^m$ by central projection. We show that the zero-sets of a suitable twist of $E$ form a family of nonclassical smooth Enriques surfaces of bidegree (4, 4) whose general member is ‘singular’ in the sense that Frobenius acts isomorphically on $H^1$, and there is a smooth divisor consisting of smooth supersingular surfaces (Frobenius acts as zero). Every nonclassical Enriques surface of bidegree (4, 4) in ${\mathbb{P}}^2\times {\mathbb{P}}^2$ that is bilinearly normal arises as a zero-set in this way.