Unobstructedness of filling secants and the Gruson-Peskine general projection theorem

TL;DR

This paper establishes unobstructed deformation properties for subvarieties intersecting fixed subvarieties, leading to smoothness and dimension results for multiple-point loci in generic projections, including from lines.

Contribution

It extends the Gruson-Peskine projection theorem by proving unobstructedness for a broader class of projections, notably from lines, and improves previous results on deformation theory.

Findings

Unobstructedness results for deformations constrained by intersections.

Smoothness and expected dimension of multiple-point loci in generic projections.

First-time results on projections from a line.

Abstract

We prove an unobstructedness result for deformations of subvarieties constrained by intersections with another, fixed subvariety. We deduce smoothness and expected-dimension results for multiple-point loci of generic projections, mainly from a point or a line, or for fibres of embedding dimension 2 or less. The current version is a substantial enhancement of the previous one, including for the first time results on projection from a line.