Strength and Hartshorne's Conjecture in high degree

TL;DR

This paper proves a special case of Hartshorne's Conjecture for high-degree subvarieties in projective space, using novel methods linked to Stillman's Conjecture, and extends field-independent results.

Contribution

It introduces a new approach connecting Hartshorne's Conjecture with ideas from Stillman's Conjecture, proving the conjecture when the dimension is much larger than the degree.

Findings

Proves Hartshorne's Conjecture in high-degree cases for large n.

Employs field-independent methods different from previous proofs.

Connects Hartshorne's Conjecture with techniques from Stillman's Conjecture.

Abstract

Hartshorne conjectured that a smooth, codimension c subvariety of n-dimensional projective space must be a complete intersection, whenever c is less than n/3. We prove this in the special case when n is much larger than the degree of the subvariety. Similar results were known in characteristic zero due to Hartshorne, Barth-Van de Ven, and others. Our proof is field independent and employs quite different methods from those previous results, as we connect Hartshorne's Conjecture with the circle of ideas initiated by Ananyan and Hochster in their proof of Stillman's Conjecture.