Geometry of genus 9 Fano 4-fold

TL;DR

This paper explores the geometry of genus 9 Fano 4-folds, constructing stable vector bundles, analyzing their line varieties, computing Chow rings, and connecting to known congruences and embeddings.

Contribution

It provides a detailed construction of stable vector bundles on genus 9 Fano 4-folds and links these to Grassmannian embeddings and known geometric structures.

Findings

Variety of lines is a hyperplane section of P1xP1xP1xP1

Computed the Chow ring revealing rich codimension 2 structure

Established links with order one congruences and Grassmannian embeddings

Abstract

References to the works of Iliev-Ranestad and Kuznetsov added. ----- In a first part we detail the construction, on a general Fano 4-fold of genus 9, of a canonical set of four stable vector bundles of rank 2, and prove that they are rigid. Those results were already known by Iliev-Ranestad and Kuznetsov with different purposes. In a second part we show that its variety of lines is an hyperplane section of P1xP1xP1xP1. Then we compute the Chow ring of a general Fano 4-fold, which appears to have a rich structure in codimension 2. The 4-bundles gives embeddings in a Grassmannian G(2,6), and the link with the order one congruence discovered by E. Mezzeti and P de Poi is done. We will also describe in this part the normalization of the non quadraticaly normal variety they constructed, and also its variety of plane cubics and detail the zak duality in this case.