A remarkable moduli space of rank 6 vector bundles related to cubic surfaces

TL;DR

This paper investigates a special moduli space of rank 6 vector bundles on projective 3-space, revealing its structure, connections to cubic surfaces, and properties of associated tensor involutions, with implications for algebraic geometry.

Contribution

It characterizes an irreducible component of the moduli space of rank 6 bundles, relates these bundles to cubic surfaces, and explores tensor involutions and their geometric significance.

Findings

The moduli space contains an irreducible open subset of dimension 19.

Constructs relate these bundles to cubic surfaces and their duals.

Studies connect tensor involutions with classical algebraic geometry concepts.

Abstract

We study the moduli space $\fM^s(6;3,6,4)$ of simple rank 6 vector bundles $\E$ on $\PP^3$ with Chern polynomial $1+3t+6t^2+4t^3$ and properties of these bundles, especially we prove some partial results concerning their stability. We first recall how these bundles are related to the construction of sextic nodal surfaces in $\PP^3$ having an even set of 56 nodes (cf. \cite{CaTo}). We prove that there is an open set, corresponding to the simple bundles with minimal cohomology, which is irreducible of dimension 19 and bimeromorphic to an open set $\fA^0$ of the G.I.T. quotient space of the projective space $\fB:=\{B\in \PP(U^\vee\otimes W\otimes V^\vee)\}$ of triple tensors of type $(3,3,4)$ by the natural action of $SL(W)\times SL(U)$. We give several constructions for these bundles, which relate them to cubic surfaces in 3-space $\PP^3$ and to cubic surfaces in the dual space $(\PP^3)^{\vee}$. One of these constructions, suggested by Igor Dolgachev, generalizes to other types of tensors. Moreover, we relate the socalled {\em cross-product involution} for $(3,3,4)$-tensors, introduced in \cite{CaTo}, with the Schur quadric associated to a cubic surface in $\PP^3$ and study further properties of this involution.