Torelli theorems for some Steiner bundles

TL;DR

This paper establishes Torelli-type theorems for Steiner bundles on projective space, showing how to recover geometric data from the bundles, and provides new proofs for existing results.

Contribution

It proves Torelli theorems for tautological bundles associated with positive divisors on very ample linear series, extending previous work and offering new proofs.

Findings

Torelli theorems are established for certain Steiner bundles.

Recovery of geometric data from Steiner bundles is possible under specified conditions.

A new proof of Dolgachev-Kapranov's result is provided.

Abstract

A Steiner bundle is a vector bundle on projective space arising as the cokernel of the map defined by a matrix of linear forms. These come up in various geometric settings, and by now they are the subject of a considerable literature. Starting with work of Dolgachev and Kapranov from 1993, several authors have considered the question of whether one can recover from the bundle the geometric data used to construct it. Here we prove such Torelli-type statements for the tautological bundles associated to sufficiently positive divisors on any very ample linear series. In an appendix, we give a new proof of the result of Dolgachev-Kapranov.