A rank-$2$ vector bundle on ${\mathbb P}^2\times {\mathbb P}^2$ and projective geometry of nonclassical Enriques surfaces in characteristic 2
Ziv Ran, Jürgen Rathmann
TL;DR
The paper constructs a rank-$2$ indecomposable vector bundle $E$ on $\\mathbb{P}^2\\times\\mathbb{P}^2$ in characteristic $2$ as the cohomology of a simple monad $0\\to O\to Q_L\otimes Q_h\to O(L+h)\\to 0$, and analyzes its cohomology and geometric zeros. The general zero-set of a suitable twist of $E$ is a smooth nonclassical Enriques surface of bidegree $(4,4)$ with $K_S=O_S$; among these, a divisor of supersingular members exists, and every bilinearly normal nonclassical Enriques surface of this bidegree arises as such a zero-set. The work interlaces monad/elements of elementary modification with Beilinson-type techniques to relate the bundle to the Enriques geometry, moduli, and embeddings into $\\mathbb{P}^2\\times\\mathbb{P}^2$. It also provides a detailed description of minimal zero-sets, a reducible model, and a moduli framework linking bundles and Enriques surfaces, together with explicit graded-module data for sections and their ideals. The results reveal a tight, characteristic-$2$ phenomenon: a new, intrinsic route to Enriques surfaces via nonliftable vector bundles on a Segre product.
Abstract
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.